Vapor-Liquid Interfaces, Bubbles and Droplets: Fundamentals and Applications

Heat and Mass Transfer Series Editors: D. Mewes and F. Mayinger For further volumes: http://www.springer.com/series/42...

Author: Shigeo Fujikawa | Takeru Yano | Masao Watanabe

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Heat and Mass Transfer Series Editors: D. Mewes and F. Mayinger

For further volumes: http://www.springer.com/series/4247

Shigeo Fujikawa · Takeru Yano · Masao Watanabe

Vapor-Liquid Interfaces, Bubbles and Droplets Fundamentals and Applications

With 74 Figures

123

Prof. Shigeo Fujikawa Hokkaido University Dept. Mechanical & Space Engineering Kita 13, Nishi 8 Sapporo 060-8628 Japan [email protected]

Prof. Takeru Yano Osaka University Dept. Mechanical Engineering Yamada-oka 2-1 Suita 565-0871 Japan [email protected]

Prof. Masao Watanabe Hokkaido University Dept. Mechanical & Space Engineering Kita 13, Nishi 8 Sapporo 060-8628 Japan [email protected]

ISSN 1860-4846 e-ISSN 1860-4854 ISBN 978-3-642-18037-8 e-ISBN 978-3-642-18038-5 DOI 10.1007/978-3-642-18038-5 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011923076 c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: eStudio Calamar S.L., Heidelberg Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

This book is the outgrowth of work done over 30 years by the first author’s group in the departments of Mechanical Engineering at Kyoto University, Mechanical Systems Engineering at Toyama Prefectural University, and Mechanical and Space Engineering at Hokkaido University. The work is concerned with basics of evaporation and condensation at the vapor–liquid interface where the bulk vapor phase and the bulk liquid phase of the same molecules coexist side by side. It focuses on physical understanding and mathematical description of interfacial phenomena in length scales ranging from a molecular size to a usual fluid-dynamic one, such as kinetic and fluid-dynamic boundary conditions including the evaporation and condensation coefficients, vapor pressure and surface tension for nanodroplets, and applications of fluid-dynamic boundary conditions to vapor bubble dynamics. The meaning and significance of subjects to be discussed in the book are described in some detail in Chap. 1. It is needless to say that the evaporation and condensation are of paramount importance in various fields of engineering, physics, chemistry, meteorology, and oceanography. As examples of current topics related to the evaporation and condensation, we can refer to flows around aircraft in clouds, bubble formation in liquid fuels of rockets, vapor explosion in nuclear reactors and volcanoes, vapor bubble formation in LNG transport process, heterogeneous reaction on droplet and aerosol surfaces in the atmosphere, and so on. The crucial point in these problems can be attributed to boundary conditions at the interface for both the Boltzmann equation and the set of Navier–Stokes equations. It was 2005 when a kinetic boundary condition (KBC) for the Boltzmann equation was formulated in a physically correct form. However, accurate values of the evaporation and condensation coefficients for any vapors have not been determined up to now, and therefore, we can not obtain still now physically correct solutions to these problems in theoretical and numerical ways. Historically, since the end of nineteen century, it has been known that the evaporation or condensation process requires the kinetic theory of gases for its analysis, and numerous investigations have been made by the kinetic approach, resulting in various fruits. However, in 1990s, it has been recognized that the kinetic theory of gases on the evaporation or condensation further needs microscopic information of molecules at the interface, e.g., correct KBC, exact values of the evaporation and condensation coefficients included in the KBC. Since then, molecular dynamics (MD) has received much v

vi

Preface

attention for simulation of the evaporation and condensation, and become a powerful tool to get microscopic information of the interface at atomic and molecular levels. The authors have engaged in investigation of the evaporation and condensation at the interface by using their unique methodology based on MD, molecular gas dynamics, and shock wave. Using MD, they have made numerical simulations of molecular motions in domains consisting of the bulk vapor of argon, its liquid, and the planar interface between them, and thereby formulated the physically correct KBC. Furthermore, using shock waves, they have made experiments of condensation for methanol and water vapors in nanometer and microsecond scales and deduced values of the evaporation and condensation coefficients of these materials by the aid of the polyatomic version of the Gaussian–BGK Boltzmann equation, a governing equation in molecular gas dynamics. The authors try to describe contents dealt with in this book as precisely as possible by restricting them to only their own work and to connect tightly them ranging from the microscopic to macroscopic scales. The evaporation or condensation phenomenon in the three space domains with utterly different length scales is analyzed by means of MD, the Gaussian–BGK Boltzmann equation, and the set of Navier– Stokes equations. Matching methods among the domains or the different governing equations are presented, and the reasonable matching between the microscopic and macroscopic scales is carried out to give the closed forms of the boundary conditions for both the Gaussian–BGK Boltzmann equation and the set of Navier–Stokes equations. A set of boundary conditions for the latter is applied to dynamics of a single vapor bubble in liquids as an application. However, the authors must say that they had to restrict the problems on the boundary conditions and the evaporation and condensation coefficients to only a single-component vapor–liquid two-phase system and to weak evaporation or condensation because of overwhelming difficulties of the problems. A two-phase system consisting of a liquid and its vapor-noncondensable gas mixture is of importance in engineering applications. However, the derivation of physically correct kinetic and fluid-dynamic boundary conditions have not been accomplished and these are under development. Problems of such a system as well as strong evaporation or condensation are left as challenging subjects in the future. The contributions to the chapters of this book are as follows: S. Fujikawa to Chaps. 1, 3, and 4; T. Yano to Chap. 2, and Appendices A and B; M. Watanabe to Chap. 5 and Appendix C. In writing this book, the authors are indebted to the following colleagues, their former Ph. D students; Prof. T. Ishiyama has contributed to Chap. 2 as his Ph. D work, Prof. K. Kobayashi to Chap. 3 as his Ph. D work, Dr. H. Yaguchi to Chap. 4 as his Ph. D work, and Drs S. Nakamura and M. Inaba partly to Chap. 3 as their Ph. D works. Without their contributions, this book might not have been born. The authors would also like to appreciate helps of Mr. Y. Nozaki, the technician in the first author’s laboratory, for his fine technique in making the experimental apparatuses, Miss K. Itagaki for typing the manuscripts, and Mrs. Y. Fujikawa for making fine figures. Finally, the first author would like to express his deepest gratitude to Ministry of Education, Culture, Sports, Science

Preface

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and Technology-Japan and Japan Society for the Promotion of Science for their continuous financial supports to his work on the evaporation and condensation over 30 years. Thanks to the financial supports, he could continue to do such a challenging work and accomplish his mission. Hokkaido, Japan Osaka, Japan Hokkaido, Japan November, 2010

Shigeo Fujikawa Takeru Yano Masao Watanabe

Contents

1 Significance of Molecular and Fluid-Dynamic Approaches to Interface Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Vapor–Liquid Interface and Kinetic Boundary Condition (KBC) . . . 1.2 Why Are Measurements of αe and αc So Difficult? . . . . . . . . . . . . . . . 1.2.1 Unsteady Nonequilibrium Condensation Induced by Shock Wave Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Temporal Transition Phenomenon of Interface Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Mechanism of Temporal Transition Phenomenon . . . . . . . . . . 1.3 Realization of Nonequilibrium States . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Another Prerequisition and Shock Wave . . . . . . . . . . . . . . . . . 1.3.2 Previous Studies of Condensation by Shock Wave . . . . . . . . . 1.4 Constitution of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Kinetic Boundary Condition at the Interface . . . . . . . . . . . . . . . . . . . . . . 2.1 Microscopic Description of Molecular Systems . . . . . . . . . . . . . . . . . . 2.1.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Liouville Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Definitions of Macroscopic Variables and Equations in Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Molecular Dynamics Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Lennard-Jones Potential and Normalization of Variables . . . . 2.2.2 Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Example: Vapor–Liquid Equilibrium State . . . . . . . . . . . . . . . 2.3 Kinetic Theory of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Boundary Condition for the Boltzmann Equation . . . . . . . . . . 2.4 Kinetic Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Evaporation into Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Evaporation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 6 6 10 11 13 13 14 15 16

19 19 21 23 24 31 31 33 35 38 39 43 45 46 49

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2.4.3

Condensation Coefficient and KBC in Weak Condensation States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Asymptotic Analysis of Weak Condensation State of Methanol . . . . 2.5.1 Problem and Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Asymptotic Analysis for Small Knudsen Numbers . . . . . . . . . 2.5.3 Boundary Condition for the Equations in Fluid-Dynamics Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Condensation Coefficient as a Linear Function of Mass Flux 2.6 Criticism on Hertz–Knudsen–Langmuir and Schrage Formulas . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Methods for the Measurement of Evaporation and Condensation Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Review of αe , αc , KBC, and Gaussian–BGK Boltzmann Equation . . 3.1.1 Definitions of αe and αc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Extension of Monatomic Version of KBC to Polyatomic One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 KBC Expressed by Net Mass Flux Measured at the Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Gaussian–BGK Boltzmann Equation in Moving Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Shock Tube Method for Measurement of Condensation Coefficient . 3.2.1 Principle of Shock Tube Method . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Characteristics of Film Condensation at Endwall behind Reflected Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Mathematical Modeling of Film Condensation on Shock Tube Endwall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Boundary Condition at Infinity in Vapor . . . . . . . . . . . . . . . . . 3.2.5 Heat Conduction in Liquid Film and Shock Tube Endwall . . 3.2.6 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Shock Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Schematic and Performance of Shock Tube . . . . . . . . . . . . . . . 3.3.2 Effect of Noncondensable Gases on Liquid Film Growth . . . 3.3.3 Effect of Association of Molecules on Vapor State . . . . . . . . . 3.4 Optical Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Theory of Optical Interferometer . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Method of Optical Data Analysis . . . . . . . . . . . . . . . . . . . . . . . 3.5 Properties of Adsorbed Liquid Film on Optical Glass Surface . . . . . . 3.5.1 Treatment of Optical Glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Thickness of Temporarily Adsorbed Liquid Film . . . . . . . . . . 3.5.3 Refractive Index of Initially Adsorbed Liquid Film . . . . . . . . 3.6 Deduction of Condensation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Typical Output Examples of Energy Reflectance . . . . . . . . . . 3.6.2 Time Changes of Liquid Film Thickness . . . . . . . . . . . . . . . . .

52 54 55 58 61 64 66 67

71 71 71 72 76 77 78 78 80 82 84 84 85 86 86 87 88 89 89 92 93 93 94 95 96 96 98

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3.6.3 3.6.4

Propagation Process of Shock Waves . . . . . . . . . . . . . . . . . . . . Time Changes of Macroscopic Quantities and Condensation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.5 Values of αe and αc for Water and Methanol . . . . . . . . . . . . . . 3.7 Sound Resonance Method for Measurement of Evaporation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Vapor Pressure, Surface Tension, and Evaporation Coefficient for Nanodroplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Significance of Molecular Dynamics Analysis for Nanodroplets . . . . 4.2 Method of MD Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Computational Method of Pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Equilibrium States of Nanodroplets and Planar Liquid Films . . . . . . . 4.4.1 General Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Density Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Pressure Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Differentiability of Normal Pressure with Respect to Radial Coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 Laplace Equation and Surface Tension . . . . . . . . . . . . . . . . . . . 4.4.6 Kelvin Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.7 Tolman Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Mass Transport Across Nanodroplet Surface . . . . . . . . . . . . . . . . . . . . 4.5.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Evaporation and Condensation Coefficients, and Mass Transfer Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Vacuum Evaporation Simulations . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Mass Fluxes and Evaporation Coefficient . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Dynamics of Spherical Vapor Bubble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Fluid-dynamic Definition of Interface . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Kinematics of Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Interface Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Interface Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Time Variation of Area of Surface Element . . . . . . . . . . . . . . . 5.2.4 Surface Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Equilibrium Thermodynamics of the Interface . . . . . . . . . . . . 5.3 General Conservation Equation at Interface . . . . . . . . . . . . . . . . . . . . . 5.3.1 Conservation Equations in Bulk Fluids . . . . . . . . . . . . . . . . . . 5.3.2 Conservation Equation in Frame Moving with Interface . . . . 5.3.3 Integration Form of Conservation Equation . . . . . . . . . . . . . . . 5.3.4 Flux Balance on Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Conservation of Mass on Interface . . . . . . . . . . . . . . . . . . . . . .

100 101 103 106 108

111 111 113 115 116 116 116 120 123 124 126 129 130 130 131 132 133 140 143 143 145 145 145 147 150 152 153 153 154 155 156 157

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5.3.6 Conservation of Momentum on Interface . . . . . . . . . . . . . . . . . 5.3.7 Conservation of Energy on Interface . . . . . . . . . . . . . . . . . . . . 5.4 Spherical Vapor Bubble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Governing Equations for Spherical Bubble . . . . . . . . . . . . . . . 5.4.2 Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Practical Description of Bubble Motion . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Flow Fields in Liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Uniform Pressure in Bubble Interior . . . . . . . . . . . . . . . . . . . . . 5.5.3 Temperature, Pressure, and Velocity Fields . . . . . . . . . . . . . . . 5.5.4 Boundary Conditions of Temperature Field . . . . . . . . . . . . . . . 5.6 Temperature Field of Bubble Exterior . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Lagrangian Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Transformation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 Laplace Transform of Heat Equation . . . . . . . . . . . . . . . . . . . . 5.6.4 Inverse Laplace Transform of Heat Equation . . . . . . . . . . . . . 5.6.5 Liquid Temperature at Bubble Wall . . . . . . . . . . . . . . . . . . . . . 5.6.6 Gradient of Liquid Temperature at Bubble Wall . . . . . . . . . . . 5.7 Temperature Field of Bubble Interior . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Adiabatic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2 Lagrangian Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.3 Boundary Layer Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.4 Solution of Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.5 Pressure and Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Structure of Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Bubble Expansion with Uniform Interior . . . . . . . . . . . . . . . . . . . . . . . 5.9.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.2 Governing Equations and Conditions . . . . . . . . . . . . . . . . . . . . 5.9.3 Heat Equation for Liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.4 Solution of Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.5 Asymptotic Growth of Vapor Bubble . . . . . . . . . . . . . . . . . . . . 5.9.6 Bubble Motion Coupled with Heat Conduction . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

159 161 162 163 165 168 171 172 172 174 175 176 176 177 179 181 186 188 189 190 191 191 193 196 197 199 199 200 202 203 206 208 209

Appendix A Vectors, Tensors, and Their Notations . . . . . . . . . . . . . . . . . . 211 A.1 Scalar, Vector, and Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 A.2 Einstein Summation Convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Appendix B Equations in Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 215 B.1 Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 B.2 Conservation Equations in Component Forms . . . . . . . . . . . . . . . . . . . 218

Contents

Appendix C Supplements to Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1 Generalized Stokes Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Characteristic Time of Heat Conduction . . . . . . . . . . . . . . . . . . . . . . . . C.3 Abel’s Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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219 219 221 223

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

Chapter 1

Significance of Molecular and Fluid-Dynamic Approaches to Interface Phenomena

Abstract In this chapter, we introduce the fundamentals of the planar vapor–liquid interface between the bulk vapor phase and the bulk liquid phase of the same molecules, stressing some key concepts such as the transition layer between them, the Knudsen layer near the interface in the vapor region, and the boundary conditions at the interface for the Boltzmann equation and the set of Navier–Stokes equation. The reason why measurements of the evaporation and condensation coefficients in the boundary conditions have been difficult is clarified in a theoretical way. The significance of the matching among different governing dynamics, i.e., molecular dynamics (MD), molecular gas dynamics, and fluid dynamics for vapor flows near the interface is discussed to make relations among the following chapters clear.

1.1 Vapor–Liquid Interface and Kinetic Boundary Condition (KBC) In fluid dynamics and molecular gas dynamics, boundary conditions are of paramount importance because they have relevance to the drag and lift exerted on bodies, and heat and mass transport across boundaries. Especially, the boundary conditions for the interface of the bulk vapor phase and the bulk liquid phase, at which evaporation or condensation occurs, involve some difficult problems [4, 11]. This is because the derivation of the boundary conditions requires detailed information of molecular phenomena at the interface, while the governing equations such as the set of Navier–Stokes equations in fluid dynamics and the Boltzmann equation in molecular gas dynamics can be derived from macroscopic and microscopic conservation laws, respectively.1 In fact, recent studies on the boundary conditions at the interface have made significant progress by molecular dynamics (MD) simulations [8, 12, 16, 18, 24–26].

1

The set of Navier–Stokes equations is summarized in Appendix B at the end of this book, and the Boltzmann equation is discussed in Sect. 2.3.

S. Fujikawa et al., Vapor-Liquid Interfaces, Bubbles and Droplets, Heat and Mass C Springer-Verlag Berlin Heidelberg 2011 Transfer, DOI 10.1007/978-3-642-18038-5_1,

1

2

1 Significance of Molecular and Fluid-Dynamic Approaches

ρ (kg/m3)

1000

(a = 4, b = 4) (a = 2, b = 2)

100

(a = 1, b = 1)

10 1 −2

−1

0

1 z∗

2

3

4

Fig. 1.1 Profiles of averaged density for argon at 85 K for some cases of (a, b). The dashed line denotes the saturated vapor density (ρV = 4.59 kg/m3 ) at 85 K

When the evaporation or condensation exists in the interface, the vapor near the interface is in a nonequilibrium state in the sense that the velocity distribution function of molecules deviates from the Maxwellian (the Maxwell distribution function) prescribed by a temperature of the interface,2 as will be discussed in Chap. 2. Let us first discuss the interface in a molecular level. Figure 1.1 shows profiles of averaged density ρ numerically obtained from MD simulations of argon [18]. As can be seen, the density continuously varies between the bulk liquid density ρ L and the bulk vapor one ρV . The region where the density changes is called the (density) transition layer. The parameters a and b in the figure represent the deviation from the equilibrium state. The equilibrium state corresponds to a = b = 1, where ρ V = 4.59 kg/m3 (the saturated vapor density at 85 K), and ρ L = 1410 kg/m3 . The vacuum evaporation state [16] is realized when a = 0. For a = b = 2 and 4, the vapors have higher densities than the saturated vapor density and negative velocities, which means net condensation states. Note that the compression factor p/(ρ RT ) is confirmed to be nearly unity in all cases, and hence the vapor can be regarded as an ideal gas. When the net condensation occurs, the interface moves toward the vapor phase. We therefore introduce a moving coordinate system [16], z ∗ = [z − (Z m − vs t)]/δ and vs = Js /ρ L , where Z m and δ are respectively the center position on a fixed coordinate and the 10–90 thickness (= 0.63 nm) of the transition layer, vs is the speed of the moving coordinate, t is the time from the beginning of MD simulations, and Js is the nonaveraged net mass flux across the interface. We can see that the profiles are almost flat in the range 2 < z ∗ < 4 of width 2δ in spite of the fact that the vapor is not in a local equilibrium state. This suggests that molecular collisions√rarely happen there. In fact, the Knudsen number estimated by Kn = /(2δ) = 1/[ 2π dm2 (ρ/m)2δ] is large (dm is the diameter of a molecule, m 2

According to molecular gas dynamics [29], an equilibrium state between the bulk vapor phase and the bulk liquid phase is defined as the state in which the velocity distribution function f of vapor molecules is given by the stationary Maxwellian in the coordinate system fixed at the vapor–liquid interface. Details are discussed in Sect. 2.3.1.

1.1

Vapor–Liquid Interface and Kinetic Boundary Condition (KBC)

Molecular Dynamics

Molecular Gas Dynamics

Matching

3

Fluid Dynamics

Matching

Vapor

Liquid Vapor–Liquid Interface Kinetic Boundary Condition Evaporation Coefficient? Condensation Coefficient?

Mass Momentum Energy

Transport Process?

Boundary Conditions?

Fig. 1.2 The figure shows the whole space to be considered in this book. The space consists of the bulk liquid phase, the transition layer, the planar vapor–liquid interface, the Knudsen layer, and the bulk vapor phase of the same molecules in turn from the left-hand side. The space can be classified into three regions as follows: the transition region, the nonequilibrium region, and the local equilibrium region. The three regions obey different governing equations, i.e., molecular dynamics (MD) in the transition region, the Boltzmann equation in the nonequilibrium region, and the set of Navier–Stokes equations in the local equilibrium region. Open circles represent molecules, but they are not figured in the local equilibrium region because the fluid is assumed to be there continuum. The figure is symbolically depicted in largely different scales for the three regions

is the mass of a molecule, and is the mean free path of vapor molecules);3 if dm is replaced by the parameter σ (= 0.341 nm for argon) in the Lennard-Jones 12-6 potential,4 Kn = 20.9, 13.5, and 7.3 for a = b = 1, 2, and 4, respectively. Since the thickness δ of the transition layer is regarded as zero in the kinetic theory and the change in the vapor condition in the range 2 < z ∗ < 4 is negligible, the interface may be defined at an arbitrary position in this range. That is, the interface locates in the vapor phase adjacent to the vapor-side edge of the transition layer. We call it the kinetic interface. Hereafter, the kinetic interface will be called just the interface. As shown in Fig. 1.2, there exists a nonequilibrium region in the neighborhood of the interface in the vapor region and it is called the Knudsen layer. The extent of this layer is of the order of the mean free path of vapor molecules. The nonequilibrium behavior of the vapor in the Knudsen layer plays an important role in the evaporation or condensation. Vapor flows accompanied with the evaporation or condensation across the interface should therefore be treated by molecular gas dynamics based on the Boltzmann equation [4, 28, 29]. The Boltzmann equation then requires the kinetic boundary condition (KBC) which prescribes the velocity distribution of molecules leaving the interface for the vapor phase. When the evaporation or condensation across the interface is weak, the KBC is expressed by the product of the two-dimensional Gauss distribution with mean zero and variance RTT for the tangential components of molecular velocity and the one-dimensional Gauss distribution with mean zero and variance RTL , as will be 3

√ The mean free path is given by = m/( 2π dm2 ρ), Eq. (2.72) in Sect. 2.3.1.

4

See Sect. 2.2.1.

4

1 Significance of Molecular and Fluid-Dynamic Approaches

explained in Chap. 2 [18]; the temperature TT is a linear function of energy flux across the interface and TL is the temperature of liquid. For the weak evaporation or condensation, TT can be regarded to be approximately equal to TL . The KBC has also a factor including the well-defined evaporation coefficient αe and condensation coefficient αc ; αe is identical with αc in the equilibrium state. This KBC reduces to the conventional KBC in the limit of the equilibrium state, i.e., TT = TL , but it does not contain any arbitrary parameter unlike the conventional KBC. The authors should note that any KBC for an arbitrarily strong evaporation or condensation has not been derived so far and its formulation is a challenging future work. There has been a long history over αe and αc since pioneering studies of Hertz [15] and Kundsen [19]. For reference, values of αe and αc of water vapor are shown in Table 1.1 for the αe -values and Table 1.2 for the αc -values; these tables are reproduced on the basis of data of αe and αc reported in Marek and Straub’s paper [23]. The recent MD simulation has succeeded in the determination of

Year

Table 1.1 The evaporation coefficient αe of water Author(s) αe

Temperature(◦ C)

1925 1931 1931 1933 1935 1939 1940 1953 1954 1954 1955 1959 1964 1964 1965 1967 1969 1971 1971 1971 1971 1973 1975 1975 1975 1975 1976

Rideal Alty Alty and Nicoll Alty Alty and Mackay Baranaev Pr¨uger Hammecke and Kappler Hickman Hickman and Torpey Kappler Fuchs Campbell Delaney et al. Mendelson and Yerazunis Maa Maa Cammenga et al. Cammenga et al. Duguid and Stampfer Tamir and Hasson Levine Davies et al. Kochurova et al. Narusawa and Springer Narusawa and Springer Bonacci et al.

25–30 5.9–32 12.1 −7.5–25 10.3–32.6 10–50 100 20 5.9–7.3 1.2 3.8–20.2 20 44.6–83.0 −0.8–4.1 38.9–78.3 0.05 0.8 24–30 18 25–35 42–50 20–28 4–19.5 25.5–34.5 18–27 18–27 2.1–8.7

1978 1987 1989 2005

Barnes ˘ Cukanov Hagen et al. Ishiyama et al. (MD)

0.0037–0.0042 0.0083–0.0155 0.0156 0.0289–0.0584 0.0061–0.0392 0.033–0.034 0.02 0.045 0.254–0.532 0.0047 0.0992–0.1015 0.03–0.034 0.0014–0.0122 0.0336–0.0545 0.0008–0.0038 1 1 0.002 0.248–0.380 0.5–1 0.10–0.30 1 1 0.050–0.065 0.038 0.19 0.065–0.665 (avg.0.54) 0.0002 0.008–0.034 0.13 1

25 39.8 16 ≈ 36

1.1

Vapor–Liquid Interface and Kinetic Boundary Condition (KBC)

5

Year

Table 1.2 The condensation coefficient αc of water vapor Author(s) αc

Temperature(◦ C)

1961 1963 1963 1964 1964 1965 1965 1967 1968 1969 1969 1971 1971 1973 1974 1974 1975 1975 1976 1976 1976 1978 1986 1987 1989 2010

Berman Nabavian and Bromley Wakeshima and Takata Goldstein Jamieson Jamieson Tanner et al. Mills and Seban Tanner et al. Maa Wenzel Magal Tamir and Hasson Vietti and Schuster Chodes et al. Gollub et al. Sinnarwalla et al. Vietti and Fastook Vietti and Fastook Bonacci et al. Finkelstein and Tamir Neizvestnyj et al. Hatamiya and Tanaka Garnier et al. Hagen et al. Fujikawa et al.

10 7–50 −16.1–5.1 25–30 0–70 – 100 7.6–10.2 22–46 0.8–8.2 22–46 25.9–82.8 48.5–105.5 – 23.9–24.9 11.4–17.5 22.5–25.7 20.8–23.2 20 5.5–7.0 60–99 20 6.9–26.9 ≈ 20–25 16 17–20

1 0.35–1 0.015–0.020 ≈ 0.1 0.305 0.35 > 0.08 0.45–1 > 0.1 1 1.0 0.040–0.044 0.09–0.35 0.21 0.031–0.037 0.010–0.012 0.021–0.032 1 0.1–1 0.417–0.693 0.006–0.060 0.3–1 0.2–0.6 0.01 0.01 1

αe -values for water [17] as shown in Table 1.1, although the simulation result is not yet verified by experiments. Concerning αc , it had not been determined accurately before the authors’ values in Table 1.2.5 We can see that the values of αe and αc largely scatter in the range of more than one hundred times. In the next section, we will consider theoretically the reason why the determination of the values of αe and αc has been so difficult. Without reliable αe and αc -values, the KBC remains open for ever. The establishment of the KBC allows us to derive a set of boundary conditions for the set of Navier–Stokes equations, i.e., a set of continuity, momentum, and energy equations in the local equilibrium region, fluid-dynamics region outside the Knudsen layer, by theoretical analysis of the Knudsen layer based on the Gaussian– BGK Boltzmann equation [1], as will be discussed in Chap. 2. The Gaussian–BGK Boltzmann equation is the only polyatomic version of the Boltzmann equation satisfying the H-theorem. For resolving the above-mentioned problems of αe , αc , and the boundary conditions for the set of Navier–Stokes equations, we have at present no any consistent law or any consistent system of governing equation. We have

5

Fujikawa et al.’s experimental values are given in Fig. 3.24 in Sect. 3.6.5.

6

1 Significance of Molecular and Fluid-Dynamic Approaches

the only theoretical method and two kinds of governing equations, i.e., MD, the Gaussian–BGK Boltzmann equation, and the set of Navier–Stokes equations. In this book, matching among MD, the Gaussian–BGK Boltzmann equation, and the set of Navier–Stokes equations will be consistently done.

1.2 Why Are Measurements of αe and αc So Difficult? In this section, we demonstrate that the difficulty in the measurement of αe and αc lies in the existence of different time scales essential for the phase change phenomena. This difficulty can be overcome by conducting the measurement of the condensation induced by an abrupt pressure elevation caused by the reflection of shock wave at the interface. We therefore start with the discussion of a shock reflection phenomenon.

1.2.1 Unsteady Nonequilibrium Condensation Induced by Shock Wave Reflection As mentioned in Sect. 1.1, the αe and αc -values measured in the past largely scatter in the range of more than one hundred times. We will here discuss the reason why these values are so different in such a wide range. The determination of αe or αc must be made through the measurement of a small amount of net mass flux of the evaporation or condensation at the interface in a nonequilibrium state; the measurement is not feasible at the equilibrium state. Such a nonequilibrium state can be realized by the following way. Let us consider the situation where the half-infinite extent of a vapor is in contact with the half-infinite extent of the liquid phase of the vapor, and these are facing each other with the plane interface between and in an equilibrium state. As shown in Fig. 1.3, a shock wave advancing from the right-hand side in the vapor collides with the interface and it is reflected, and propagating in the right-hand direction as the time elapses; the time is running upward. Just at the instant when the shock wave is reflected at the interface, the pressure, temperature, and density of the vapor increase stepwise from the initially low state to a high one. The temperature of the vapor at the interface changes little because of the large difference in heat capacities of the vapor and liquid. The Knudsen layer is formed near the interface in the vapor, and the thermal boundary layer also develops outside the Knudsen layer with the lapse of time. The vapor pressure at the interface then becomes higher than the saturated vapor pressure at the interfacial liquid temperature. As a result, the vapor becomes supersaturated at the interface, consequently condensing, and the interface moves toward right-hand side with time. The net mass flux of condensation at the interface, which we need, can be obtained from the measurement of interface movement. This problem has been solved by Fujikawa et al. [10] for the system of shock tube endwall, liquid film, and vapor on the basis of the method of matched asymptotic expansions, as will be mentioned in Sect. 3.2.2;

1.2

Why Are Measurements of αe and αc So Difficult?

7

Thermal Boundary Layer

Liquid

Vapor Flow

Interface

Time

Particle Path Trajectory of Reflected Shock Wave

O

Trajectory of Incident Shock Wave Vapor

Fig. 1.3 The propagation process of the shock wave in the vapor advancing toward and reflecting from the liquid surface. The time is running upward

the reflection of a shock wave at the shock tube endwall in a noncondensable gas has been analyzed by Clarke [5]. The problem shown in Fig. 1.3 is a simplified version of Ref. [10] and the result of its analysis can be summarized as follows. For simplicity, we will assume α = αe = αc and adopt a set of fluid-dynamic boundary conditions at the interface for the set of Navier–Stokes equations as follows [1, 28, 29]6 : p − p∗ 1 u = , √ √ ∗ ∗ p 2RTL C4 − 2 π 1−α α

(1.1)

T − TL u = d4∗ √ , TL 2RTL

(1.2)

where TL is the liquid temperature at the interface, T is the vapor temperature at the interface, p ∗ is the saturated vapor pressure at TL , p is the vapor pressure at the interface, u is the vapor velocity at the interface, R is the gas constant per unit mass, and C4∗ = −2.13204 and d4∗ = −0.44675 for Boltzmann–Krook–Welander (BKW) model; C4∗ and d4∗ are slightly different among models such as BKW and

6 This assumption holds for the weak evaporation or condensation which takes place near the equilibrium state, as will be discussed in Chaps. 2 and 3. Equations (1.1) and (1.2) are respectively Eqs. (2.138) and (2.139) in Sect. 2.5.3 for the case that the vapor flow is one-dimensional and the flow velocity is much larger than the moving velocity of the interface. And, also see Footnote 20 in Sect. 2.5.3.

8

1 Significance of Molecular and Fluid-Dynamic Approaches

hard-sphere models.7 The more general boundary conditions for polyatomic gases will be given in Chap. 2 in this book, but Eqs. (1.1) and (1.2) suffice for the present purpose. Fujikawa et al. have solved the set of one-dimensional Navier–Stokes equations of vapor and the heat conduction equation of lquid together with Eqs. (1.1) and (1.2) by the method of matched asymptotic expansions [9, 10]. The time-dependent position δ(t) of the interface from its initial one is then described at the level of the first approximation by dδ(t) = dt

√

2RTL (t) ρ∞ T∞ p∞ − p∗ (TL ) , φ(α) ρ L TL (t) p ∗ (TL )

ρL L TL (t) = T0 + kL

DL π

0

t

dδ(t˜)/dt˜ dt˜. t − t˜

(1.3)

(1.4)

Here, t is the time measured from the instant of the step change of the state, T0 is the initial temperature of the vapor and liquid, ρ∞ and T∞ are the vapor density and temperature far from the interface behind the reflected shock wave, ρ L is the liquid density, D L is the thermal diffusivity of liquid, k L is the thermal conductivity of liquid, L is the latent heat of condensation, and φ(α) and p∗ (TL ) are respectively given by √ 1−α , φ(α) = −C4∗ + 2 π α ∗

p (TL ) =

p0∗

(1.5)

A 1 − A + TL (t) , T0

(1.6)

where p0∗ = p ∗ (T0 ) and A = bT0 /(c + T0 )2 in which b and c are given later [Eqs. (1.9) and (1.10)]; ρ L , D L , k L , and L are constant values. Equation (1.6) is obtained from Antoine’s equation [30] by Taylor’s expansion, and p∗ (T0 ) = p0∗ as it should be so; Antoine’s equation is given below in Eq. (1.7). The saturated vapor pressure p ∗ and the saturated vapor density ρ ∗ can be obtained by Antoine’s equation and the state equation for ideal gases: p = exp a − ∗

ρ∗ =

p∗ , RTL

b c + TL

,

(1.7)

(1.8)

7 The values of C ∗ and d ∗ are given in Eq. (2.134) for hard-sphere gas and Eq. (2.136) for the 4 4 BKW model.

1.2

Why Are Measurements of αe and αc So Difficult?

9

where the units of TL and p∗ are respectively (K) and (Pa), and the constant values a, b, and c are for methanol vapor a = 23.4803,

b = 3626.55,

c = −34.29,

(1.9)

b = 3816.44,

c = −46.13.

(1.10)

and for water vapor a = 23.1964,

From Eqs. (1.3), (1.4), and (1.6), we obtain a Volterra integral equation of the second kind on the displacement speed of the interface as follows: dδ(t) β2 = β1 − √ dt π

t

0

dδ(t˜)/dt˜ dt˜, t − t˜

(1.11)

where p∞ ( p∞ − p0∗ ) β1 = φ(α)ρ L p0∗

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

2 , RT0

p∞ ( p∞ − p0∗ + 2Ap∞ )L β2 = φ(α)k L p0∗

⎪ ⎪ ⎪ ⎪ . ⎪ 3 ⎭ 2RT

(1.12)

DL

0

The solution of Eq. (1.11) can be obtained as [6] √ dδ(t) = β1 exp(β22 t)erfc(β2 t), dt

(1.13)

where √ 2 erfc(β2 t) = √ π

∞ √

β2 t

e−x dx. 2

(1.14)

Integrating Eq. (1.13) with respect to the time t leads to √ √ β1 2 2 δ(t) = 2 exp(β2 t)erfc(β2 t) + √ β2 t − 1 . π β2

(1.15)

where δ(0) = 0. From Eqs. (1.4), (1.11), and (1.13), we obtain TL (t) = T0 +

√ √ β1 ρ L L D L 1 − exp(β22 t)erfc(β2 t) . β2 k L

(1.16)

where TL (0) = T0 . The position and temperature of the interface are respectively described by Eqs. (1.15) and (1.16).

10

1 Significance of Molecular and Fluid-Dynamic Approaches

1.2.2 Temporal Transition Phenomenon of Interface Displacement Equations (1.15) √ and (1.16) can be classified into two cases depending on values of the variable β2 t as follows: √ (1) for β2 t 18 ; δ(t) = β1 t,

(1.17)

2β1 ρ L L TL (t) = T0 + kL

DL t, π

(1.18)

√ (2) for β2 t 19 ; β1 2 β1 √ t − 2, δ(t) = √ π β2 β2

(1.19)

√ ρ L L D L β1 (= const.), TL = T0 + kL β2

(1.20)

where we should notice, for the following discussion, that β1 and β1 /β22 are dependent on φ(α), i.e., α, while β1 /β2 is independent of φ(α). The displacement of the interface is drastically influenced by√φ(α) and the time measured from the instant of in the step change of the state. For β2 t 1, the position of the interface changes √ proportion to the time and the interface speed depends on φ(α), while for β2 t 1 the position changes in proportion to the square root of the time, and its change gradually becomes independent of φ(α) as the time lapse and becomes strongly dependent on thermophysical properties of the vapor and liquid. This suggests that the position of the interface should be measured in early time stages just after the

8

For small values of x, erfc x can be expressed as follows [3]: ∞ 2 (−1)n x 2n+1 , erfc x = 1 − erf x = 1 − √ (2n + 1)n! π n=0

where erf x is the error function. 9

For large values of x, erfc x can be expressed as follows [3]: √ ∞ 1 1 1 1.3 π 1.3 . . . (2n − 3) 2 2 erfc x = e−ξ dξ = e−x − 3 + 2 5 − · · · + (−1)n−1 2 2 x 2x 2 x 2n−1 x 2n−1 x

+ (−1)n

1.3 . . . (2n − 1) 2n

∞ x

e−ξ dξ . ξ 2n 2

1.2

Why Are Measurements of αe and αc So Difficult?

11

step change caused by the shock reflection, when we determine α through the measurement of the interface displacement. It is quite natural to notice, in the above discussion, that there exists √ a transition time between the t-proportion displacement of the interface and the t-proportion √ one and that this time can be deduced from the relation β2 t = O(1). Defining the √ transition time as τt when β2 τt = 1, we obtain 2RT03 τt = DL

φ(α)k L p0∗ p∞ ( p∞ − p0∗ + 2Ap∞ )L

2 .

(1.21)

The transition time τt is in proportion to [φ(α)]2 . Generally, φ(α) approaches −C4∗ (= 2.13204 for hard-sphere gas) as α does unity, and on the other hand, φ(α) approaches infinity as α does zero. Evaluating τt for small pressure changes by 5 % from the saturated vapor pressures at 290 K for methanol and water vapors, we obtain, e.g., for α = 1, τt ∼ = 0.1 μs for methanol vapor and τt ∼ = 7 μs for water vapor, respectively. In both cases, the transition times are very short. The reason why the transition time of methanol vapor is shorter than that of water vapor is principally because the saturated vapor pressure of methanol is six times higher than that of water at 290 K; the vapor with the higher saturated pressure causes the more mass flux and the more rapid temperature rise of liquid, thereby resulting in the shorter transition time. If the liquid is a very thin film and it is on a solid wall with a thermal conductivity higher than that of the liquid, the transition time becomes a little longer (see Sect. 3.2.2). Although the restriction of transition time for the measurement of interface displacement is not so strict, we can understand that the measurement should be carried out in the time scale of microseconds, not in time scale of milliseconds or seconds. All values of αe and αc shown in Tables 1.1 and 1.2 have been measured in the past through the displacement of plane or curved liquid surface directly, or other indirect ways. However, there was no recognition of the existence of the temporal transition phenomenon in the past measurements. This is one of reasons why the values of αe and αc measured have been largely different.

1.2.3 Mechanism of Temporal Transition Phenomenon As clarified in the preceding subsection, the displacement of the interface greatly depends on the change of the saturated vapor pressure at the interfacial liquid temperature. The time evolution of the temperature is given by Eq. (1.18) for t τt and by Eq. (1.20) for t τt . The temperature at the time τt is given by √ β1 ρ L L D L TL = T0 + 0.5724 . β2 k L

(1.22)

12

1 Significance of Molecular and Fluid-Dynamic Approaches

At this time stage, the temperature rise is about √ 57% of the temperature variation from its initial value to the asymptote [= β1 ρ L L D L /(β2 k L )]. Therefore, we can understand that the transition time is the characteristic time when the temperature approaches the asymptote over the time. Now, let us consider the balance of heat fluxes at the interface in order to understand the mechanism of the temporal transition phenomenon. The balance equation of heat fluxes per unit time and unit interface area is given by ρL L

⎫ dδ(t) ∼ ⎪ = heat conduction into liquid ⎪ ⎪ ⎬ dt ⎪ ⎪ ⎪ ⎭

TL (t) − T0 √ , ∝ t

√ ρLL DL β1 kL β2

TL

(1.23)

T0

Interface Time Lapse Liquid Space Coordinate

Vapor O

(a) Before Transition Time

√ ρLL DL β1 kL β2

TL

T0

Interface Time Lapse

Liquid Space Coordinate

Vapor O

(b) After Transition Time

Fig. 1.4 Temperature changes of liquid surface and liquid interior: (a) before transition time and (b) after transition time

1.3

Realization of Nonequilibrium States

13

√ because the thermal boundary layer in the liquid develops according to t; the heat flux term due to heat conduction in the vapor in the left-hand side of Eq. (1.23) is neglected because it is very small compared with that due to condensation, ρ L Ldδ(t)/dt [10]. For t τt , from Eqs. (1.18) and (1.23), we obtain dδ(t)/dt = const., i.e., δ(t) ∝ t. It can therefore be concluded that the thermal process before τt is the one in which the interface temperature increases by condensation heat and the heat diffuses into the liquid interior, as shown in Fig. 1.4a. After τt , on the other hand, the interface temperature approaches the asymptote, and in consequence, from √ √ Eqs. (1.20) and (1.23), we obtain dδ(t)/dt ∝ 1/ t, i.e., δ(t) ∝ t. At this time stage, the temperature approaches the asymptote, and the heat diffuses into the liquid interior, as shown in Fig. 1.4b. Such a process of condensation after τt may be called a thermal diffusion-controlled condensation. If the measurement of α is conducted based upon the time history of delta for a long range of thermal diffusion-controlled condensation, the determined α-values easily become inaccurate. Furthermore, the determination of α in the time scale of τt is extremely difficult because of the lack of a suitable method for realization of nonequilibrium state of a vapor at the interface, except for our shock wave method. The lack of both recognition of the existence of transition time and suitable method for realization of the nonequilibrium state is the reason why the values of αe and αc have largely scattered as shown in Tables 1.1 and 1.2.

1.3 Realization of Nonequilibrium States 1.3.1 Another Prerequisition and Shock Wave In the preceding section, it has been elucidated that the displacement of the interface should be measured in the time scale of the transition time defined by Eq. (1.21), when the evaporation or condensation coefficient, i.e., αe or αc , is determined from the measurement of the net mass flux of evaporation or condensation. This is one prerequisition for determination of αe and αc . There is another prerequisition for preparation of a nonequilibrium state of a vapor. The nonequilibrium state must be realized in a time scale much shorter than the transition time, and the realization in such a short time is almost impossible by any mechanical tools. The method by shock wave discussed in the preceding section may also be the best one from the following view points: (i) The nonequilibrium state of a vapor can be realized at a vapor–liquid interface in the time scale of the mean free time of vapor molecules, when the shock wave advancing toward the liquid surface in the vapor is reflected at the surface. (ii) The nonequilibirium state can be held for a sufficiently long time compared with the transition time.

14

1 Significance of Molecular and Fluid-Dynamic Approaches

The mean free time τm is given by 1 τm = 4π(ρ/m)dm2

πm , kT

(1.24)

where k is the Boltzmann constant (k = 1.3806504 × 10−23 J/K) and T is the vapor temperature. Evaluating the values of τm for the saturated methanol and water vapors at 290 K, we obtain τm ∼ = 2 ns for methanol and τm ∼ = 8 ns for water; the values of dm for the vapors are replaced by σ00 in Ref. [17]. These values are much smaller than those of τt for both vapors, as estimated in Sect. 1.2.2. Let us here summarize the prerequisitions important for the determination of αe and αc : (i) The nonequilibrium state of the vapor must be realized in the time scale of nanoseconds. (ii) The determination of αe and αc must be performed in the time scale of microseconds. The shock wave is the useful tool satisfying these prerequisitions. Technical details of the method based on shock wave will be given in Chap. 3.

1.3.2 Previous Studies of Condensation by Shock Wave Goldstein [13] first made an experiment on condensation of water vapor on the sidewall of a shock tube and tried to measure the condensation coefficient. The condensation was produced in the following way. Behind an incident shock wave, the vapor is compressed and heated, but it is rapidly cooled because of the large difference in heat capacities between the vapor and the shock tube sidewall. An unsteady thermal boundary layer forms on the wall and it develops toward the vapor region. Under an appropriate initial condition, the vapor becomes supersaturated on the wall surface and it begins to condense on the wall. The experiment showed that filmwise or dropwise condensation took place on the sidewall depending upon the nature of wall surface. Following Goldstein, Grosse and Smith [14] also confirmed the condensation of Freon-11 vapor on the sidewall of a shock tube. The growth of a liquid film formed on the wall was, however, found to be disturbed by the presence of the viscous boundary layer on the wall behind the incident shock wave. Smith [27] improved Goldstein’s method by paying attention to film condensation on the endwall of a shock tube behind a reflected shock wave. The experiment demonstrated that a liquid film formed uniformly on the endwall and that its growth behavior might be analyzed in terms of two different models, one for time less than 10 µs and one for longer times. Except for the early stage of about 10 µs after the reflection of the shock wave, the liquid film grew in proportion to the square root of the time for time

1.4

Constitution of This Book

15

intervals of the order of milliseconds. Unfortunately, Smith was not able to deduce the condensation coefficient from the data obtained because he had no theoretical tool for its deduction. Maerefat et al. [21, 22] directed by Fujikawa followed the experimental method which Smith proposed [27], and succeeded in determining the condensation coefficients of methanol, water, and carbon tetrachloride vapors, by combining both experimental data of the liquid film growth and their theoretical analysis of the set of Naiver–Stokes equations. Later, Fujikawa et al. [7, 11] and Kobayashi et al. [20] adopted the Gaussian–BGK Boltzmann equation instead of the set of Navier–Stokes equations to analyze the reflection process of the shock wave at the shock tube endwall and the subsequent growth of the liquid film. The Gaussian–BGK Boltzmann equation was solved in a fully numerical way under the new KBC formulated with MD simulations [18]; in the analysis, the growth rate of the liquid film formed on the shock tube endwall was incorporated into the KBC, and this made it possible to fuse the experimental data and numerical ones and to deduce further microscopic information on the condensation. The most recent results of the evaporation and condensation coefficients for methanol and water vapors will be presented in Chap. 3.

1.4 Constitution of This Book In Chap. 2, we shall formulate the physically correct KBC for the Boltzmann equation on the basis of MD simulations. We demonstrate that when the evaporation or condensation across the interface is weak, the KBC is expressed by the product of a three-dimensional Gauss distribution function and a factor including the evaporation coefficient αe and condensation coefficient αc defined well. The Gaussian–BGK Boltzmann equation and the theoretical analysis of the Knudsen layer near the interface will be treated. We will derive the boundary conditions for the set of Navier– Stokes equations, i.e., the continuity, momentum, and energy equations in the fluid dynamics region outside the Knudsen layer based on the Gaussian–BGK Boltzmann equation and the KBC. In Chap. 3, the KBC formulated in Chap. 2 is extended to the case of polyatomic molecules by inclusion of the distribution function for energy associated with the internal structure of molecules. We determine αe and αc for methanol and water vapors by making use of carefully prepared shock tube experiments and precise nonequilibrium molecular gas dynamics simulations. This will lead to the substantial termination of the long lasting controversy over αe and αc . A method based on sound resonance is also presented for exact determination of αe as a complementary tool to the shock tube. In Chap. 4, we shall further extend the MD simulations conducted in Chap. 2 to nanodroplets and elucidate effects of droplet size on the surface tension, the equilibrium vapor pressure, and αe of the nanodroplets, and also clarify the applicable limit of thermodynamics in the nanodroplets. In Chap. 5, we shall derive a set of equations describing the dynamics of a spherical vapor bubble accompanied by the evaporation or condensation at the interface as an example of the application of the physically correct sets of equations

16

1 Significance of Molecular and Fluid-Dynamic Approaches

to practically important problems. The analytical method presented enables us to solve in a mathematically rigorous manner the flow fields of both internal and external the bubble, and the dynamics at the bubble wall by taking temperature and density distributions inside the bubble into account under the assumption of uniform pressure.

References 1. P. Andries, P.L. Tallec, J.P. Perlat, B. Perthame, The Gaussian-BGK model of Boltzmann equation with small Prandtl number. Eur. J. Mech. B-Fluids 19, 813–830 (2000) 2. K. Aoki, Y. Sone, Gas flows around the condensed phase with strong evaporation or condensation: Fluid dynamic equation and its boundary condition on the interface and their application, in Advances in Kinetic Theory and Continuum Mechanics, eds. by R. Gatignol, Soubbaramayer (Springer, Berlin, 1991), pp. 43–54 3. H.S. Carslaw, J.C.Jaeger, Conduction of Heat in Solids, 2nd edn. (Oxford University Press, Oxford, 1959) 4. C. Cercignani, Rarefied Gas Dynamics (Cambridge University Press, New York, NY 2000) 5. J.F. Clarke, The reflexion of a plane shockwave from a heat-conducting wall. Proc. R. Soc. Lond. A 299, 221–237 (1967) 6. A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Tables of Integral Transforms, Vol. I. (McGraw-Hill, New York, NY, 1954) 7. S. Fujikawa, Molecular transport phenomena and the kinetic boundary condition at the vaporliquid interface, in Proceedings of 7th World Conference Experimental Heat Transfer, Fluid Mechanics and Thermodynamics, eds. by J.S. Szmyd, J. Spalek, T.A. Kowalewski (Jagiellonian University, Krakow, 2009), pp. 81–97 8. A. Frezzotti, P. Grosfils, S. Toxvaerd, Evidence of an inverted temperature gradient during evaporation/condensation of a Lennard-Jones fluid. Phys. Fluids 15, 2837–2842 (2003) 9. S. Fujikawa, M. Okuda, T. Akamatsu, T. Goto, Non-equilibrium vapour condensation on a shock-tube endwall behind a reflected shock wave. J. Fluid Mech. 183, 293–324 (1987) 10. S. Fujikawa, M. Kotani, N. Takasugi, Theory of film condensation on shock-tube endwall behind reflected shock wave: Theoretical basis for determination of condensation coefficient. JSME Int. J. 40, 159–165 (1997) 11. S. Fujikawa, Molecular gas dynamics applied to phase change processes at a vapor-liquid interface: Shock-tube experiment and MGD computation for methanol, in Proceedings of the 7th World Conference Experimental Heat Transfer, Fluid Mechanics and Thermodynamics, eds. by J.S. Szmyd, J. Spalek, T.A. Kowalewski (Jagiellonian University, Krakow, 2009), pp. 81–97 12. A. Gilde, N. Siladke, C.P. Lawrence, Molecular dynamics simulations of water transport through butanol films. J. Phys. Chem. A 113, 8586–8590 (2009) 13. R. Goldstein, Study of water vapor condensation on shock-tube walls. J. Chem. Phys. 40, 2793–2799 (1964) 14. F.A. Grosse, W.R. Smith, Vapor condensation in a shock tube: Electrostatic effects. Phys. Fluids 11, 735–739 (1968) 15. H. Hertz, Ueber die Verdunstung der Flüssigkeiten, insbesondere des Quecksilbers, im luftleeren Raume. Ann. Phys. Chemie 17, 177–200 (1882) 16. T. Ishiyama, T. Yano, S. Fujikawa, Molecular dynamics study of kinetic boundary condition at an interface between argon vapor and its condensed phase. Phys. Fluids 16, 2899–2906 (2004) 17. T. Ishiyama, T. Yano, S. Fujikawa, Molecular dynamics study of kinetic boundary condition at an interface between polyatomic vapor and its condensed phase. Phys. Fluids 16, 4713–4726 (2004)

References

17

18. T. Ishiyama, T. Yano, S. Fujikawa, Kinetic boundary condition at a vapor-liquid interface. Phys. Rev. Lett. 95, 084504 (2005) 19. M. Knudsen, Die MaximaleVerdampfunggeschwindigkeit des Quecksilbers. Ann. Phys. Chemie 47, 697–708 (1915) 20. K. Kobayashi, S. Watanabe, D. Yamano, T. Yano, S. Fujikawa, Condensation coefficient of water in a weak condensation state. Fluid Dyn. Res. 40, 585–596 (2008) 21. M. Maerefat, T. Akamatsu, S. Fujikawa, Non-equilibrium condensation of water and carbontetrachloride vapour in a shock-tube. Exp. Fluids 9, 345–351 (1990) 22. M. Maerefat, S. Fujikawa, T. Akamatsu, T. Goto, T. Mizutani, An experimental study of non-equilibrium vapour condensation in a shock-tube. Exp. Fluids 7, 513–520 (1989) 23. R. Marek, J. Straub, Analysis of the evaporation coefficient and the condensation coefficient of water. Int. J. Heat Mass Transf. 44, 39–53 (2001) 24. R. Meland, A. Frezzotti, T. Ytrehus, B. Hafskjold, Nonequilibrium molecular-dynamics simulation of net evaporation and net condensation, and evaluation of the gas-kinetic boundary condition at the interface. Phys. Fluids 16, 223–243 (2004) 25. A. Morita, M. Sugiyama, H. Kameda, S. Koda, D.R. Hanson, Mass accommodation coefficient of water: Molecular dynamics simulation and revised analysis of droplet train/flow reactor experiment. J. Phys. Chem. B 108, 9111–9120 (2004) 26. G. Nagayama, T. Tsuruta, A general expression for the condensation coefficient based on transition state theory and molecular dynamics simulation. J. Chem. Phys. 118, 1392–1399 (2003) 27. W.R. Smith, Vapor-liquid condensation in a shock tube, in Proceedings of the 9th International. Shock Tube Symposium, eds. by B. Bershader, W. Griffith (Stanford University, Stanford, CA, 1973), pp. 785–792 28. Y. Sone, Kinetic Theory and Fluid Dynamics (Birkhäuser, Boston, MA, 2002) 29. Y. Sone, Molecular Gas Dynamics (Birkhäuser, Boston, MA, 2007) 30. The Society of Chemical Engineers of Japan (ed.), Handbook of Chemical Engineering (Maruzen, Tokyo, 1988), pp. 18–27

Chapter 2

Kinetic Boundary Condition at the Interface

Abstract The vapor–liquid interface can exist only where the bulk vapor phase and the bulk liquid phase of the same molecules coexist side by side. Therefore, all the properties of the interface are inevitably affected by the bulk liquid and vapor phases, and vice versa. The relation among these three constituents still remains unresolved in general nonequilibrium states. However, at least in a weak nonequilibrium state, the relations can be simplified and reformulated into a form of Kinetic Boundary Condition (KBC) at the vapor–liquid interface. In this chapter, from the microscopic point of view, we explain how the two bulk phases of vapor and liquid are connected via the KBC at the interface. The main tools used here are the nonequilibrium molecular dynamics simulation of vapor–liquid two-phase system and the Boltzmann equation for vapor. Our aims in this chapter are to establish the KBC at the interface by the molecular dynamics simulation and to reduce it into the boundary condition for the vapor flows in the fluid-dynamics region outside the Knudsen layer on the interface by the asymptotic analysis of the boundary-value problem of the Boltzmann equation for small Knudsen numbers.

2.1 Microscopic Description of Molecular Systems The physical properties of materials consisted of a large number of molecules are resulted from some kinds of averages over a large number of molecules, because our utilization activities of materials are usually carried out in some scales considerably large compared with molecular scales. For example, the density of a fluid is always evaluated as an averaged mass of a number of molecules in a volume divided by the volume. Consider the measurement of the temperature of a fluid by a thermometer. The motion of molecules forming the thermometer is in an equilibrium state as a result of the energy exchange with molecules in the fluid that contacts the thermometer. The total kinetic energy of molecules forming the thermometer is then translated into the temperature of thermometer, which is equal to the temperature of the fluid in the equilibrium state. In translating the kinetic energy to the temperature, we employ a fundamental relation in statistical mechanics [30]

S. Fujikawa et al., Vapor-Liquid Interfaces, Bubbles and Droplets, Heat and Mass C Springer-Verlag Berlin Heidelberg 2011 Transfer, DOI 10.1007/978-3-642-18038-5_2,

19

20

2 Kinetic Boundary Condition at the Interface

3 1 kT = m(ξx2 + ξ y2 + ξz2 ) , 2 2

(2.1)

where k is the Boltzmann constant, T is the temperature in the equilibrium state, m is the mass of a molecule, and (ξx , ξ y , ξz ) is the molecular velocity.1 For the case of temperature measurement, the angle brackets · · · in the right-hand side of Eq. (2.1) means the average over the molecules forming the thermometer and also a time average for some time duration of reading the scale of the thermometer. Thus, the macroscopic variables, e.g., T in the left-hand side of Eq. (2.1), are defined by some kinds of averages of microscopic variables, e.g., the molecular velocity (ξx , ξ y , ξz ) in the right-hand side of Eq. (2.1). When the temperature and pressure of a material (liquids or gases) concerned are uniform over its some volume and the materials is at rest, the relations connecting the temperature, pressure, density, internal energy, and so on can be described without using the microscopic information. In the last century, such relations have been compiled and integrated into thermodynamics, which explains the relations among macroscopic variables of materials in equilibrium states and does not contain the microscopic information at least apparently. Although fluid dynamics can discuss behaviors of nonuniform and flowing liquids and gases, its foundation is supported by thermodynamics under the assumption of the local equilibrium, which requires that the fluid in a sufficiently small volume2 is locally in an equilibrium state,3 even if the temperature and pressure are not uniform and the fluid is flowing over large scales. The interface, however, is a thin layer by its definition, and thermodynamics (and fluid dynamics) in such a thin layer has not been established yet or may not be expected to be established. Therefore, we have to consider the vapor–liquid interface and its neighborhood from the microscopic point of view. For example, in Sect. 4.4, using molecular dynamics simulations, we demonstrate that when a spherical nanodroplet and the surrounding vapor are in an equilibrium state in the sense of statistical mechanics, the thermodynamical vapor–liquid equilibrium condition (an equality of chemical potentials of vapor and liquid) does not hold. We start with a brief explanation of a general microscopic description of molecular systems.

1 For

simplicity, the fluid and the thermometer are assumed to be composed of monatomic molecules and at rest in the macroscopic sense. The formula (m/2) ξi2 = kT /2 (i = x, y, z) is called the equipartition theorem or the law of equipartition of energy.

2 The

sufficiently small volume in fluid dynamics is sufficiently large in molecular scales so that it may contain a number of molecules. Thus, the macroscopic variables defined by some kinds of averages can be regarded as continuous functions of the space coordinates and the time. If the fluid is an ideal gas in the standard state, the number of molecules in a cube with a side-length 1 µm is 2.6867774×107 . The number of molecules per unit volume is called the Loschmidt constant.

3 The

actualization of local equilibrium requires a sufficient number of molecular interactions (intermolecular collisions).

2.1

Microscopic Description of Molecular Systems

21

2.1.1 Equation of Motion For simplicity, we deal with a single component system consisted of a large number of monatomic molecules, assuming that the motion of each molecule is determined by classical mechanics (without any quantum effects), the molecules are electrically neutral, and any types of association and dissociation do not occur. Then, Newton’s equation of motion is the starting point: m

d2 x (i) = f (i) , dt 2

(i = 1, 2, . . . , N ),

(2.2)

where m is the mass of a molecule, N is the total number of molecules in the system concerned, x (i) is the position vector of the ith molecule, t is the time, and f (i) is the force exerted on the ith molecule. Equation (2.2) can be written as (i)

m (i)

d2 x j

(i)

where x j and f j

dt 2

(i)

= fj ,

(i = 1, 2, . . . , N ; j = 1, 2, 3),

(2.3)

are the jth components of vectors x (i) and f (i) , respectively.

Once given an explicit form of the force f (i) and initial positions and velocities of all molecules, Newton’s equation of motion (2.2) can in principle be solved, e.g., numercally. All the macroscopic properties of materials can then be determined through their definitions in terms of some kinds of averages of the microscopic quantities, i.e., x (i) , dx (i) /dt, and d2 x (i) /dt 2 (or f (i) ) for all molecules; the definitions of macroscopic variables, such as the temperature, density, velocity, pressure, internal energy, and so on, are shown in Sects. 2.1.3, 2.3.1, and 2.5.1. In general, the force f (i) includes external forces such as the gravity. The magnitude of acceleration due to gravity is, however, negligibly small compared with a typical intermolecular force in molecular scales in time and space, and hence we only consider f (i) in Eq. (2.2) as the force acting between molecules, i.e., the intermolecular force.4 Then, it seems to be plausible to assume that the force f (i) is a conservative force, and its potential U is a function of intermolecular distances only. We further limit ourselves to the case that the potential U may be approximated as a pairwise and additive one, i.e., 1 φ(rik ), 2 N

U=

N

(2.4)

i=1 k=1 k=i

where rik = x (i) − x (k) is the distance between the ith molecule and the kth molecule (i = k) and φ is a function of intermolecular distance rik . The factor 1/2

4 Forces

acting between atoms making up a molecule (covalent bonds, ionic bonds, and metallic bonds) are called the intramolecular forces.

22

2 Kinetic Boundary Condition at the Interface

in the right-hand side of Eq. (2.4) is introduced since the double sum with respect to i and k counts each pair twice. The force exerted on the ith molecule can then be defined by f (i) = −

∂U , ∂ x (i)

(2.5)

and expressed in a component form as f j(i)

=−

N ∂rik (i)

k=1 k=i

∂x j

φ (rik ) = −

N x (i) − x (k) j j

rik

k=1 k=i

φ (rik ),

(2.6)

or in a vector form as f (i) = −

N x (i) − x (k) φ (rik ), rik

(2.7)

k=1 k=i

where φ (rik ) denotes the derivative of φ with respect to rik . The pairwise and additive potentials are widely used in molecular dynamics (MD) simulations of liquids and gases [1, 11], although the complete validation has not been given. On the other hand, multi-body potentials are used for various MD simulations of crystalline materials such as graphite, diamond, and carbon nanotube [6]. (i) Introducing the generalized momenta p j conjugate to the generalized coordi(i)

nates q j , Newton’s equation of motion (2.3) with Eq. (2.6) can be reformulated to the equations of motion in Hamilton’s form, or Hamilton’s canonical equations of motion [22]: dq (i) j dt

=

∂H

d p(i) j

∂pj

dt

, (i)

=−

∂H (i)

∂q j

Here,

,

(i = 1, 2, . . . , N ; j = 1, 2, 3).

(i)

(i)

pj = m

dx j dt

,

(i)

(i)

qj = xj ,

H=

n=1

⎢1 ⎢ ⎣2

(2.9) ⎤

⎡ N

(2.8)

3 k=1

(n) (n) pk pk

m

+

1 2

N k=1 k=n

⎥ φ(rnk )⎥ ⎦,

(2.10)

where H is the Hamiltonian of the whole system. Note that we confine ourselves to the molecular system of N monatomic molecules, and hence the kinetic energy included in the Hamiltonian H is that of translational motions only [the inclusion of

2.1

Microscopic Description of Molecular Systems

23

the internal rotational motions of a polyatomic molecule into Eqs. (2.8), (2.9), and (2.10) is straightforward]. Since the Hamiltonian H defined by Eq. (2.10) does not depend explicitly on the time t, it is a global constant of the motion and we write H = E, where E is the total energy of the system.

2.1.2 Liouville Equation An arbitrary state of the N molecular system can be specified by using the 6N dimensional phase space (q, p), where q = (q1(1) , q2(1), q3(1), q1(2), . . . , q2(N ), q3(N ) ) and p = ( p1(1), p2(1), p3(1), p1(2), . . . , p2(N ), p3(N ) ) [22, 30, 32]. The results derived from the concept of the phase space should be the same as those obtained from the solutions of Newton’s equation of motion (2.3) with Eq. (2.6), while the former is much more suitable for the deduction of macroscopic properties from the microscopic information, because the macroscopic properties are associated with some kinds of averages over a number of molecules. The 6N -dimensional phase space is sometimes called the Γ -space. Specifying an arbitrary point (q, p) in the 6N -dimensional phase space at a given time t is equivalent to specifying a set of initial conditions of Newton’s equations of motion for N molecules. From the existence and uniqueness of the solution of initial-value problem of a set of Newton’s equations of motion (a set of ordinary differential equations), there exists a trajectory of solution that passes through the specified arbitrary point (q, p) in the phase space at the time. That is, the phase space is filled with the trajectories of solutions of a set of Newton’s equations of motion and a point in the phase space represents a state of the N molecular system. The density distribution function (probability density function) F(q, p, t) in the phase space is now defined by dP = F(q, p, t) dq d p,

(2.11)

where dP is the probability that a point (q, p), moving in the phase space according to Newton’s equation of motion, lies in a 6N -dimensional volume element dq d p = (1) (1) (N ) (1) (1) (N ) dq1 dq2 · · · dq3 d p1 d p2 · · · d p3 at a time t. Thus, the probability of finding the system of N molecules in a region χ in the phase space is P(χ ) =

χ

F(q, p, t) dq d p,

(2.12)

where the integration is taken over the region χ . If χ is the whole 6N -dimensional space, then P(χ ) = 1. The expected value of an arbitrary function G(q, p) is given by G(q, p)F(q, p, t) dq d p.

(2.13)

24

2 Kinetic Boundary Condition at the Interface

From the conservation of probability, in the same manner as the derivation of equation of mass continuity in fluid dynamics, we have [22, 30, 32] N 3 dq (i) d p (i) ∂ dF ∂F ∂ j j F = + + (i) F = 0, (i) dt ∂t dt dt ∂pj i=1 j=1 ∂q j

(2.14)

where dF/dt is the total time derivative of F. Expanding the derivatives with respect (i) to q (i) j and p j in Eq. (2.14) and applying the equations of motion in Hamilton’s form (2.8), we can transform Eq. (2.14) into (i) 3 N d p (i) ∂ F dq j ∂ F ∂F j + + = 0. ∂t dt ∂q (i) dt ∂ p (i) i=1 j=1 j j

(2.15)

Equation (2.15) is called the Liouville equation. Using Eq. (2.9) and Newton’s equation of motion (2.3) in Eq. (2.15), we have ∂F + ∂t N

3

(i)

pj

∂F

m ∂q (i) j

i=1 j=1

+

f j(i)

∂F ∂ p (i) j

= 0.

(2.16)

Equation (2.16) is also called the Liouville equation. If an isolated system is in an equilibrium state, then the probability density F must be time independent and we have N 3 i=1 j=1

(i)

pj

∂F

m ∂q (i) j

+

(i) fj

∂F (i)

∂pj

= 0.

(2.17)

This means that the flow in the phase space is steady and incompressible in the sense of fluid dynamics [22, 30, 32]. In general nonequilibrium states, Eq. (2.17) does not hold and we have to return to the Liouville equation (2.16).

2.1.3 Definitions of Macroscopic Variables and Equations in Fluid Dynamics The formal relations between microscopic and macroscopic variables can readily be derived from the Liouville equation (2.16) by the method of Irving and Kirkwood [15]. The method is superior in that the definitions of macroscopic variables can be applied to those in nonequilibrium states for both liquids and gases, and the resulting definitions are suitable for the use in the analysis of the data obtained by MD simulations. Note that if we restrict ourselves to the case that the fluid is an ideal gas, the definitions of macroscopic variables have been established in the kinetic

2.1

Microscopic Description of Molecular Systems

25

theory of gases (molecular gas dynamics) in terms not of the probability density F but of the velocity distribution function of gas molecules [35]. The kinetic theory of gases are summarized in Sect. 2.3. In the following, we derive the equations in fluid dynamics (conservation equations of mass, momentum, and energy) from the Liouville equation (2.16) by the method of Irving and Kirkwood [15] with a small modification for the later use in Sect. 2.2. In the original paper [15], the Dirac delta function is used instead of a scalar function χ defined below. To begin with, we define an averaged fluid density as ρ(x, t) =

N m χ (q (i), x; h)F(q, p, t) dq d p, h3

(2.18)

i=1

where the integration is taken over the whole 6N -dimensional phase space and χ (q (i), x; h) =

⎧ ⎨1

(i) q j − x j h/2 for j = 1, 2, 3,

⎩ 0 otherwise.

(2.19)

That is, if the ith molecule is in a cube with a side-length h centered at x, then χ = 1 and the integration yields the expected value that one will find the ith molecule in the cube (Fig. 2.1). Therefore, the right-hand side of Eq. (2.18) is the expected value of the number of molecules in the cube multiplied by m/ h 3 , which is the averaged fluid density ρ. Note that the introduction of the function χ is a coarse-graining process. Although the side-length h of the cube is arbitrary, it should be small compared with some length scale that is to be resolved. Similarly, an averaged fluid momentum per unit volume and an averaged total energy of fluid per unit volume are defined as

χ (q (i) , x ; h) ˙

.

˙

q (k) q (j)

x ˙ q ()

h

h

h

O Fig. 2.1 The value of the scalar function χ(q (i), x; h) is equal to unity if the ith molecule is contained in a cube with a side-length h centered at x

26

2 Kinetic Boundary Condition at the Interface

ρ(x, t)v j (x, t) =

N 1 χ (q (i), x, h) p (i) j F(q, p, t) dq d p, h3

(2.20)

i=1

N 1 ρ(x, t)E(x, t) = 3 χ (q (i), x, h)e(i) F(q, p, t) dq d p, h

(2.21)

i=1

e

(i)

(i) (i)

3 N 1 1 pj pj + = φ(rik ), 2 m 2 j=1

(2.22)

k=1 k=i

where Eq. (2.10) is used for the definition of e(i) , the sum of the kinetic and potential energies of the ith molecule.5 Thus, the fluid velocity v j and the total energy of fluid per unit mass E are defined.6 The conservation law of mass of a fluid is expressed in a partial differential equation, which can be derived from the Liouville equation (2.16) in the following manner. Multiplying Eq. (2.16) by χ (n) = χ (q (n), x, h), we have (i) 3 N p ∂F ∂ (n) (n) j (n) (i) ∂ F χ = 0. + χ fj (χ F) + (i) ∂t m ∂q (i) ∂p i=1 j=1

j

(2.23)

j

Integrating Eq. (2.23) over the whole phase space gives ∂ ∂t

χ

(n)

F dq d p +

N 3

χ (n)

i=1 j=1

+

p (i) ∂F j dq d p m ∂q (i) j

χ (n) f j(i)

∂F ∂ p (i) j

dq d p = 0.

(2.24)

With the aid of the Gauss divergence theorem, the following equations hold for an arbitrary function G, N 3 i=1 j=1 Ωq N 3 i=1 j=1 Ω p

G(q)

G( p)

∂F (i) ∂q j

∂F (i) ∂pj

dq = − dp = −

N 3 i=1 j=1 Ωq N 3 i=1 j=1 Ω p

∂G (i)

∂q j

∂G (i)

∂pj

F dq,

(2.25)

F d p,

(2.26)

5 The potential energy of the ith molecule is defined only formally. It is the total potential energy of all molecules that has the physical meaning. 6 In

Chap. 5 and in Appendix B, the internal energy per unit mass of fluid is denoted by e.

2.1

Microscopic Description of Molecular Systems

27

if the probability density F falls off rapidly outside a bounded region Ωq in the 3N -dimensional space of q in the case of Eq. (2.25) and outside a bounded region Ω p in the 3N -dimensional space of p in the case of Eq. (2.26). Using Eqs. (2.25) and (2.26), we can rewrite Eq. (2.24) into ∂ ∂t

χ

(n)

F dq d p −

3 N

(i)

respect to

(i) pj

(n)

(i)

pj

F dq d p χ (i) m ∂q j ∂ ! (n) (i) " + χ fj F dq d p = 0. (i) ∂pj

i=1 j=1

Since χ (n) and f j

∂

(i)

(2.27) (i)

are independent of p j , the differentiation of χ (n) f j

with

vanishes, i.e., !

∂ (i)

∂pj

(i)

χ (n) f j

"

= 0.

(2.28)

From the definition of the function χ , Eq. (2.19), we have N

∂

i=1

∂q (i) j

χ

(i)

p (n) j

m

=

(n)

∂

χ

∂q (n) j

p (n) j

m

∂ =− ∂x j

χ

(n)

p (n) j

m

,

(2.29)

where in the last equality we used ∂χ (n) (n)

∂q j

=−

∂χ (n) . ∂x j

(2.30)

Substituting Eqs. (2.28) and (2.29) into Eq. (2.27) and taking the sum over all n, we obtain the equation of mass continuity in fluid dynamics, ∂ρ ∂ρv j + = 0, ∂t ∂x j 3

(2.31)

j=1

where the definitions of ρ and ρv j , Eqs. (2.18) and (2.20), are used. Equation (2.31) with the definitions of ρ and v j does not require the assumption of local equilibrium unlike in the case of fluid dynamics.7 only requirement is that ρ and v j are continuously differentiable functions of x and t. This will be satisfied by choosing h in the function χ so that the cube with a side-length h centered at x contains a large number of molecules. If the fluid is a gas, this does not warrant the local equilibrium, because the mean free path of gas molecules can be very large compared with h.

7 The

28

2 Kinetic Boundary Condition at the Interface

In the same manner as the derivation of Eq. (2.27), we can derive the equations of momentum conservation of nth molecule ∂ ∂t

(n) χ (n) pk F

+

dq d p −

N 3

∂ (i)

∂q j

i=1 j=1

∂ ∂ p (i) j

χ

p (n) p(i) j (n) k

! " χ (n) pk(n) f j(i) F dq d p

m

F dq d p

= 0,

(k = 1, 2, 3),

(2.32)

where Eq. (2.20) is used. By making use of Eqs. (2.28) and (2.30), Eq. (2.32) can be rewritten into ∂ ∂t

χ (n) pk(n) F

(n) (n) 3 p pj ∂ (n) k F dq d p χ dq d p + ∂x j m j=1 (n) − χ (n) f k F dq d p = 0, (k = 1, 2, 3).

(2.33)

The third term in Eq. (2.33) can be transformed to yield8 ∂ ∂t

χ (n) pk(n) F

(n) (n) 3 p pj ∂ (n) k F dq d p χ dq d p + ∂x j m j=1

−

3 j=1

∂ ∂x j

1 h x j − q (n) j + 2 3

χ (n) f k(n) F dq d p = 0,

(k = 1, 2, 3). (2.34)

Taking the sum over n, we should have the conservation equation of the momentum of the fluid per unit volume, 3 $ ∂ρvk ∂ # ρvk v j + Pk j = 0, + ∂t ∂x j

(k = 1, 2, 3),

(2.35)

j=1

where the microscopic definition of the stress tensor Pk j is given as

In the case that the fluid is a liquid, this may be a sufficient condition for the local equilibrium, because the mean free path of liquid molecules is usually comparable with or less than a typical diameter of a molecule. 8 The factor 1 in Eq. (2.34) is an ideal limit of N → ∞ in equilibrium states. The stress tensor in 3 the original paper [15] is different from this and much more cumbersome.

2.1

Microscopic Description of Molecular Systems N 1 Pk j = 3 h

h 1 (n) (n) χ (n) F dq d p − xj − qj + fk 3 2

(n) (n)

pk p j m

n=1

29

− ρv j vk ,

( j, k = 1, 2, 3).

(2.36)

If the fluid is in a uniform equilibrium state at rest, then v j = 0 and Pk j = Pδk j , where δk j is the Kronecker delta and pressure P is given by 3 N 1 P= 3 3h

n=1 k=1

pk(n) pk(n) h (n) (n) χ (n) F dq d p. (2.37) − x k − qk + fk m 2

The energy conservation equation for the nth molecule, the counterpart of Eq. (2.34), is ∂ ∂t

χ (n) e(n) F dq d p +

(n) 3 pj ∂ F dq d p χ (n) e(n) ∂x j m j=1

−

(n) 3 p ∂ h (n) (n) j x j − q (n) + χ f F dq d p = 0. j ∂x j 2 m j

(2.38)

j=1

After taking the sum over n, the conservation equation of the total energy of the fluid per unit volume and the microscopic definition of the heat flux Q j are obtained as ∂ρ E ∂ + ∂t ∂x j 3

ρ Ev j +

j=1

3

Pi j vi + Q j

= 0,

(2.39)

i=1

(n) N pj 1 h (n) (n) (n) Qj = 3 χ (n) F dq d p fj e − xj − qj + h 2 m n=1

− ρ Ev j +

3

Pi j vi ,

( j = 1, 2, 3).

(2.40)

i=1

It may be instructive to compare the above results with the definitions of macroscopic variables in the kinetic theory of gases [35]. The definitions of the stress tensor and the heat flux in the kinetic theory of gases are given as follows: Pi j = =

(ξi − vi )(ξ j − v j ) f (x, ξ , t) dξ ξi ξ j f (x, ξ , t) dξ − ρvi v j ,

(i, j, = 1, 2, 3),

(2.41)

30

2 Kinetic Boundary Condition at the Interface

1 (ξi − vi )2 f (x, ξ , t) dξ (ξ j − v j ) 2 i=1 3 3 3 3 1 2 1 2 = ξj ξi f (x, ξ , t) dξ − ρ vi v j + pi j vi , RT + 2 2 2 3

Qj =

i=1

i=1

i=1

( j = 1, 2, 3), (2.42) where ξi is the ith component of molecular velocity, T is the temperature, R = k/m is the gas constant (per unit mass), the integration is taken over the 3-dimensional space of molecular velocity ξ , and f (x, ξ , t) is the velocity distribution function of gas molecules, which gives the gas density ρ=

f (x, ξ , t) dξ .

(2.43)

The precise definition of the velocity distribution function f (x, ξ , t) is given in Sect. 2.3.1. Equations (2.41) and (2.42) are also applicable to general nonequilibrium states with the definition of temperature 3 ρ RT = 2

1 2 1 2 ξi f dξ − ρ vi . 2 2 3

3

i=1

i=1

(2.44)

As compared with Eqs. (2.36) and (2.40), one can see that the contribution of intermolecular force is neglected in Eqs. (2.41) and (2.42). Thus, the microscopic definitions of the stress tensor and the heat flux are extended to nonequilibrium states in fluids including liquids where the intermolecular force cannot be neglected. They are used for the analysis of results from MD simulations. For the definition of temperature, 3 kT = 2 N

i=1

χ (q (i) − x; h)

(i) (i) 3 3 1 pj pj m 2 vi , (2.45) F(q, p, t) dq d p − 2 m 2 j=1

i=1

is used in MD simulations, instead of Eq. (2.44). We have derived the equations for conservation laws of macroscopic variables, Eqs. (2.31), (2.35), and (2.39), with the definitions of macroscopic variables, Eqs. (2.18), (2.20), (2.21), (2.36), (2.40), and (2.45). They can be applied to general nonequilibrium states of liquids and gases, for which we cannot expect (i) the thermodynamic relations (or the assumption of local equilibrium), (ii) the stress tensor of the Newtonian fluid, and (iii) the heat flux based on the Fourier law. For example, it is well known that the stress tensor for a slightly rarefied gas contains terms related to temperature gradient, called the thermal stress [35], which does not appear in equations of fluid dynamics (see Appendix B at the end of this book).

2.2

Molecular Dynamics Simulation

31

Since our target is the vapor–liquid interface, we have to know the behavior of molecules in the liquid phase. For this purpose, we use the MD simulations for the vapor–liquid two phase system.

2.2 Molecular Dynamics Simulation In the preceding subsection, we have given the definitions of macroscopic variables in terms of the microscopic information. They can be applied to liquids and gases in general nonequilibrium states, and expressed in appropriate forms for the analysis in MD simulation. Now, we move on to the explanation of the method of MD simulations. The standard method of MD simulation numerically solves Newton’s equation of motion (2.3) for a number of molecules confined in a simulation box fixed in a physical coordinate system. The total number of molecules in the box is unchanged during the simulation if an appropriate boundary condition on the surface of the box is imposed. Such a method of simulation is called N V E simulation because the number of molecules, N , the volume of the system, V , and the total energy of molecules, E, are constant in the simulation except for numerical errors (mainly truncation errors) in the total energy. It is possible and sometimes preferred to perform simulations with a constant temperature (N V T simulation) or those with constant temperature and pressure (N P T simulation) [1, 13]. However, N V T simulation and N P T simulation solve some dynamical systems different from Newton’s equation of motion (2.3) and the Liouville equation (2.16). The differences resulted from the differences from Newton’s equation of motion have not been figured out in nonequilibrium MD simulations.9 In this book, we concentrate on the dynamics of molecules based on Newton’s equation of motion (2.3), although not restricted to N V E simulations.

2.2.1 Lennard-Jones Potential and Normalization of Variables In MD simulations, the most widely used pairwise additive potential is the LennardJones 12-6 potential [23], 1 φ(rik ), U= 2 N

N

i=1 k=1 k=i

φ(rik ) = 4

σ rik

12

−

σ rik

6 ,

(2.46)

where (J) and σ (m) are the energy and length parameters of Lennard-Jones 12-6 potential. Using these two parameters and the molecular mass m, Newton’s equation of motion (2.3) can be nondimensionalized as 9 The

macroscopic properties in equilibrium states should not be different by the difference of simulation methods.

32

2 Kinetic Boundary Condition at the Interface (i)

d2 xˆ j

dtˆ2

(i) = fˆj ,

(2.47)

where xˆ (i) j

=

x (i) j σ

,

t tˆ = √ , σ m/

fˆj(i) =

f j(i) /σ

.

(2.48)

The nondimensional quantities are signified by ˆ. Then, from Eq. (2.6), the nondimensionalized intermolecular force fˆj(i) is given by (i) fˆj = −

N xˆ (i) − xˆ (k) j j k=1 k=i

rˆik

φˆ (ˆrik ),

(2.49)

where ˆ rik ) = 4 φ(ˆ

1 rˆik

12

1 − rˆik

6 ,

rˆik = xˆ (i) − xˆ (k) .

(2.50)

The macroscopic variables are also nondimensionalized as follows: σ 3 m 1/2 Q j . 3/2 (2.51) ˆ r ) and its In Fig. 2.2, the nondimensionalized Lennard-Jones 12-6 potential φ(ˆ derivative with respect to the argument, φˆ (ˆr ), are shown in a solid curve and dashed ˆ r ) has a minimum at rˆ = 21/6 , at which a strong curve, respectively. The potential φ(ˆ 1/6 repulsive force (ˆr < 2 ) is switched to an attractive force (ˆr > 21/6 ). The repulsive force behaves like rˆ −13 and the asymptotic form of the tail of attractive force is rˆ −7 as rˆ → ∞. Figure 2.2 suggests that rˆ = 1 (r = σ ) is a reasonable measure of the diameter of the molecule modeled by the Lennard-Jones 12-6 potential. The depth ˆ corresponds to φ = and indicates of potential well, i.e., the minimum of φ, the strength of molecular interaction. Thanks to a number of simulation studies up to now, the Lennard-Jones parameters and σ suitable for modeling simple molecules such as Ar, Ne, Kr, N, O have been tabulated; for example, (/k, σ ) = (119.8 K, 0.341 nm) for Ar, (47.0 K, 0.272 nm) for Ne, (164.0 K, 0.383 nm) for Kr, (37.3 K, 0.331 nm) for N, (61.6 K, 0.295 nm) for O [1], where the Boltzmann constant k = 1.3806504 × 10−23 J/K. Before proceeding to subsections for numerical method, we note that the nondimensionalization of variables in Eqs. (2.48), (2.49), (2.50), and (2.51) are defined by the quantities in molecular scales. Effects of much larger scales in space and ρˆ =

σ 3ρ , m

kT Tˆ = ,

vj vˆ j = √ , /m

Pˆi j =

σ 3 Pi j ,

Qˆ j =

2.2

Molecular Dynamics Simulation

33

Normalized Potential & Force

3

ˆˆ dφ(r) ˆ dr

2

1

ˆˆ φ(r) 1

0

2

1/6

–1 0

1

2

3

Normalized Intermolecular Distance

ˆ r ) (solid curve) and its derivaFig. 2.2 The nondimensionalized Lennard-Jones 12-6 potential φ(ˆ tive with respect to the argument φˆ (ˆr ) (dashed curve)

time may therefore slip through standard numerical methods usually used in MD simulations.10 We focus on molecular phenomena in molecular scales.

2.2.2 Finite Difference Method Newton’s equation of motion (2.47) is a set of ordinary differential equations. Although we have many sophisticated numerical techniques for solving ordinary differential equations [29], MD simulations usually use rather simple finite difference methods. This is because we have to deal with a number of molecules in the system N , and/or we have to continue computations for quite large steps M. The number of molecules N and the number of steps M sometimes exceed N = 106 and M = 108 . Large N and M directly result in the increase in the computational time. We therefore prefer numerical methods as simple as possible with less degradation in accuracy. The methods commonly used are the leap-frog scheme, the velocity Verlet scheme, and Gear’s predictor–corrector algorithms [1, 11]. We here explain the leap-frog scheme. In the leap-frog scheme, the nondimensionalized Newton’s equation of motion (2.47) is discretized as (i) ˆ ˆ ˆ ˆ (i) ˆ 1 ˆ ˆ3 xˆ (i) j (t + t ) = xˆ j (t ) + t vˆ j (t + 2 t ) + O(t ),

10 For

example, it is natural that large-scale fluid flows are affected by the gravity.

(2.52)

34

2 Kinetic Boundary Condition at the Interface (i) (i) (i) vˆ j (tˆ + 12 tˆ ) = vˆ j (tˆ − 12 tˆ ) + tˆ fˆj (tˆ ) + O(tˆ 3 ),

(2.53)

ˆ where vˆ (i) j is the jth component of velocity of the ith molecule and t is the time step. By using the Taylor expansions, it is easy to derive Eqs. (2.52) and (2.53), and (i) to confirm that the local truncation errors are of the order of tˆ 3 . Note that fˆj (tˆ) in the right-hand side of Eq. (2.53) is given by fˆj(i) (tˆ) = −

N xˆ (i) (tˆ )− xˆ (k) (tˆ ) j j k=1 k=i

rˆik (tˆ )

φˆ (ˆrik (tˆ )),

rˆik (tˆ ) = xˆ (i) (tˆ )− xˆ (k) (tˆ ),

(2.54)

(2.55)

(i) (i) ˆ 1 ˆ ˆ 1 ˆ ˆ and hence vˆ (i) j (t + 2 t ) can be obtained by Eq. (2.53) if vˆ j (t − 2 t ) and xˆ j (t )

(i) 1 ˆ 1 ˆ ˆ ˆ are known. After obtaining vˆ (i) j (t + 2 t ), we can determine xˆ j (t + 2 t ) by Eq. (2.52), and the computation can be continued to the next time step. Thus, the time series of positions and velocities obtained by the leap-frog scheme (2.52) and (2.53) are shifted by the half of the time step. In many cases, it is important and useful to monitor the value of the total energy (total Hamiltonian) (2.10), which can be written in the nondimensional form as follows:

(i) $ 1 # (i) ˆ tˆ ) = 1 φˆ rˆik (tˆ ) . vˆ j (tˆ )vˆ j (tˆ ) + H( 2 2 N

3

i=1 j=1

N

N

(2.56)

i=1 k=1 k=i

ˆ tˆ ), we can use For the evaluation of H( (i)

vˆ j (tˆ ) =

1 (i) (i) vˆ j (tˆ + 12 tˆ ) + vˆ j (tˆ − 12 tˆ ) + O(tˆ 2 ). 2

(2.57)

ˆ tˆ ) by the use of Eq. (2.57) may not The error of the order of tˆ 2 brought into H( ˆ lead to serious problems, because Eq. (2.57) is used only in the evaluation of H, and significant errors are caused by the accumulation of local truncation errors for a large number of computational steps. Even if the number of molecules in the system is not so large, e.g., N = 104 , (i) the exact evaluation of fˆj is a hard task because of the long tail of attractive force, as shown in Fig. 2.2. Since the tail of attractive force decays as rˆ −7 , it seems to be reasonable to cut it off at some distance rˆcut from the center of the ith molecule. Many authors have so far used rˆcut = 2.5 in their MD simulations. However, the simulations of vapor–liquid interface and its neighborhood are strongly affected by the details of the numerical method, such as, the size of cut-off radius rˆcut , the thickness of liquid layer, and the area of the interface [12, 26, 40]. In particular, the

2.2

Molecular Dynamics Simulation

35

Density (kg/m3) & Temperature (K)

3000

1410 kg/m3

1000 Liquid

100

Temperature

84.6 K

Vapor

Vapor

Density

4.3 kg/m3

Interface

Interface

10

1 0

5

10 15 20 25 Space Coordinate (nm)

30

Fig. 2.3 The density and temperature in a vapor–liquid equilibrium state. In the figure, the density transition layer is shown as the interface

shorter is the cut-off radius, the lower the liquid density and the higher the vapor density. According to Refs. [12, 26, 40], to suppress artificial effects due to small rˆcut , it should be larger than or at least equal to 4.4 (it corresponds to 1.5 nm for the case of argon). Usually, the simulation is performed for a specified N molecules put in a simulation box fixed in the coordinate system. We therefore impose some boundary condition at the surface of the box. The most simple one is the periodic boundary condition [1, 11], and it enables us to avoid introducing artificial boundary conditions and to conserve the number of molecules. Needless to say, the use of the periodic boundary condition does not imply the simulation of infinitely large volume at all, even in the cases of equilibrium simulation. For example, the thickness of the vapor–liquid interface (density transition layer), as shown in Fig. 2.3 in Sect. 2.2.3 and illustrated in Fig. 2.7 in Sect. 2.4.1, is known to be a logarithmically increasing function of the area of the interface (the cross section of the simulation box) [12, 26, 40]. Furthermore, if the size of the box is not sufficiently large compared with the cut-off radius, the artificial effect of periodic boundary condition spoils the result of simulations [1, 11].

2.2.3 Example: Vapor–Liquid Equilibrium State As an example of MD simulations, we present the distributions of averaged density and temperature of a vapor–liquid coexistence system including two interfaces in Fig. 2.3, where a planar liquid layer of monatomic molecules exists between two vapor phases of the same molecules, and the system is in an equilibrium state.

36

2 Kinetic Boundary Condition at the Interface

Although the simulation is performed with nondimensional variables defined in previous subsections, the dimensional density and temperature are shown in Fig. 2.3 with the use of (/k, σ ) = (119.8 K, 0.341 nm) for argon. The temperature (84.6 K) is uniform,11 the liquid density is 1410 kg/m3 , and the vapor density is 4.3 kg/m3 , which are in agreement with known values [28] with errors less than 1%. The density profile in Fig. 2.3 shows that the vapor–liquid interface has a finite thickness. The dashed line in Fig. 2.3 denotes the edge of the bulk liquid region or bulk vapor region, where we tentatively use “bulk” to indicate that the averaged density is spatially uniform. The details of the simulation method are as follows: The simulation box is L 1 × L 2 × L 3 = 90 σ × 30 σ × 30 σ nm3 , and a planar liquid layer with thickness about 7 nm is set at around x 1 = L 1 /2 as the initial condition, and the total number of molecules is N = 17280. Under the periodic boundary condition, Newton’s equations of motion (2.47), (2.48), (2.49), and (2.50) for N molecules are numerically solved by √ using the leap-frog scheme (2.52) and (2.53) with the time step tˆ = 0.0005 (σ m/tˆ = 10−15 s) and the cut-off radius rˆ√ cut = 5 (σ rˆcut = 1.7 nm). The number of total simulation steps is M = 4 × 108 (σ m/ Mtˆ = 4 × 10−7 s). The first half of M simulation steps is dedicated to a relaxation process to a vapor– liquid equilibrium, and the results shown in Fig. 2.3 are evaluated from averages of samples obtained in the second half of M steps. The evaluation of density and temperature uses Eqs. (2.18) and (2.45). Since the phenomenon considered here is one-dimensional in the macroscopic sense, the function χ defined by Eq. (2.19) should be replaced by χ 1 (q (i) − x; h) =

⎧ ⎨1 ⎩0

(i) q1 − x1 h/2, (i) q1 − x1 > h/2.

(2.58)

Accordingly, the volume of the cube h 3 in Eqs. (2.18) and (2.45) should also be replaced by h L 2 L 3 (see Fig. 2.4). The integration of any function multiplied by the probability density F(q, p, t) with respect to q and p is naturally replaced by the summation over a large number of samples obtained in the MD simulation. The errors introduced by the cut-off of the intermolecular force are illustrated in (i) Fig. 2.5a, where the error is defined by the difference of f with rˆcut from f (i) with rˆcut = 10 for some ith molecule at some instant t. From the figure, we can approximately estimate that the error of molecules in the bulk liquid phase decreases with 10−3ˆrcut /4 as rˆcut increases, and that of molecules at the edge of the bulk liquid phase decreases more slowly with 10−ˆrcut /4 . The molecular motions in the vicinity of the interface is therefore affected by the size of cut-off radius of intermolecular potential strongly. The use of a small cut-off radius results in an equilibrium system with a high vapor density and a low liquid density [12, 26, 40].

11 The

triple point temperature is 83.8 K for argon [28].

2.2

Molecular Dynamics Simulation

37

χ1(q(i) –˙x ; h) .

˙

x3 L2

q (k) x2

q (j) L3

q () x1

O

h

Fig. 2.4 The function χ 1 for spatially one-dimensional problems in the macroscopic sense

Figure 2.5b illustrates how the system size (number of total molecules N ) affects ˆ Here, the error the total Hamiltonian Hˆ defined by (2.56) and numerical errors in H. ˆ In the figure, four systems is defined by three times the standard deviation of H. with different N (4320, 17280, 69588, 286487) are compared, for which we use the same leaf-frog scheme with the same cut-off radius rˆcut = 5 and the same time step tˆ = 0.0005, and different simulation boxes: (L 1 /σ, L 2 /σ, L 3 /σ ) = (90, 15, 15) for N = 4320, (90, 30, 30) for N = 17280, (150, 60, 60) for N = 69588, (390, 120, 120) for N = 286487. In our vapor–liquid two-phase systems, almost (b)

Bulk Liquid

10–5

10

Er

69588

17280 69588 4320 105

ia n

10–4

ro

106

H a × mi (– lt 1) on

10–3

286487

5

0

1 r×

17280

al

Edge of Bulk Liquid

To t

Total Hamiltonian & Error

Error of Intermolecular Force

(a) 10–2

4320 4

–6

10

2

4

6

ˆ rcut

8

103

104

105

Number of Molecules

Fig. 2.5 (a) The difference between the intermolecular force with various cut-off rˆcut and that with rˆcut = 10 is shown as a measure of error in the intermolecular force, where the open circle denotes the force on some molecule in the bulk liquid phase and the closed circle denotes that on some molecule at the edge of the bulk liquid phase. (b) The relation between the error in the total Hamiltonian and the number of molecules in the system. The open circles are Hˆ × (−1) and the closed circles are three times the standard deviation of Hˆ × 105 . The number near the symbol denotes the total number of molecules in the system, N . The solid line is a line of slope 1 and the dashed line slope 1/2

38

2 Kinetic Boundary Condition at the Interface

80% of Hˆ is the potential energy of molecules in liquid and hence Hˆ is substantially determined by the number of molecules in the liquid layer. This is the reason why Hˆ in Fig. 2.5b is proportional to √ N . The important conclusion from Fig. 2.5b is that if Hˆ ∝ N , then the error ∝ N . Theoretically, Hˆ should be constant in N V E simulations as well as in an isolated system, and therefore, the fluctuation in Hˆ is purely numerical. Figure 2.5b suggests that the numerical error in Hˆ also is governed by the central limit theorem.12

2.3 Kinetic Theory of Gases The kinetic theory is a microscopic theory of processes in systems not in statistical equilibrium [24]. A kinetic boundary condition means a boundary condition for a kinetic equation, the governing equation of a kinetic theory, such as the Boltzmann equation [35], the Vlasov equation for plasma [9], the Enskog equation for dense gases [31]. In contrast to the well-established kinetic theory of gases, the rigorous kinetic theory of liquids is considerably complicated [5], and its formidable mathematical difficulties impede reducing and interpreting the formal solutions. The origin of difficulties is clearly apparently random multi-body interactions of molecules in liquids. On the other hand, the kinetic theory of gases based on the Boltzmann equation, which is a mainstay of this book, deals with dilute gases in the sense that the three-body interaction of gas molecules does not occur. As a result, the kinetic theory of gases permits us to obtain a number of fruits from it, although the exact kinetic theory of gases is still mathematically difficult. In the context of this book, however, the random multi-body interactions of liquid molecules are not necessarily an obstacle for our purpose. It allows us to expect that the liquid is in a local equilibrium state everywhere except for the interface because of a sufficiently rapid relaxation to equilibrium due to frequent multi-body interactions of molecules in liquids. Once the local equilibrium in a liquid is admitted, we can focus our attention to the behavior of gas molecules under the given macroscopic condition of the liquid. Strictly speaking, if the evaporation or condensation at the interface is not so weak, the assumption of local equilibrium in the liquid may not be accepted uncritically. For example, if the heat flux across the interface due to the evaporation or condensation is fairly large, the liquid near the interface may deviate from a local equilibrium state. If the liquid near the interface is not in a local equilibrium state, we cannot specify the macroscopic condition of the liquid, which is necessary to analyze the vapor flow separately by the kinetic theory of gases. In this book, we therefore confine ourselves to the case of weak evaporation/condensation state, the precise definition of which is given in Sects. 2.5.1 and 2.5.2. Although we determine the molecular motions in the liquid in a rigorous way of the MD 12 It is easy to confirm that the probability density of numerically obtained H ˆ approaches a Gaussian with the increase in N .

2.3

Kinetic Theory of Gases

39

simulation, some constraints are still imposed on the bulk of the liquid, and we do not pursue the kinetic theory of liquids.

2.3.1 Boltzmann Equation For the present, we confine ourselves to the gases of monatomic molecules. In the kinetic theory of gases, the only unknown microscopic variable is the velocity distribution function of gas molecules, f (x, ξ , t), defined by m dN = f (x, ξ , t) dx dξ ,

(2.59)

where dN in the left-hand side denotes the number of molecules in a volume element dx dξ = dx1 dx 2 dx3 dξ1 dξ2 dξ3 centered at (x, ξ ) in the 6-dimensional space of the position x and the velocity ξ of a molecule [35] (N signifies the number of molecules). The governing equation for f is the Boltzmann equation, ∂f ∂f = J ( f ), + ξj ∂t ∂x j

(2.60)

where J ( f ) in the right-hand side represents the effects of the intermolecular collisions. Here and hereafter the Einstein summation convention is used, e.g., ∂f ∂f = ξj , ∂x j ∂x j 3

ξj

(2.61)

j=1

(see Appendix A at the end of this book). Furthermore, the notation of the jth component of a vector a, a j , will be used without notice. Once given the velocity distribution function f (x, ξ , t), the macroscopic variables are evaluated by 1 1 (2.62) ξi f dξ , T = (ξ j − v j )2 f dξ , ρ= f dξ , vi = ρ 3ρ R where ρ is the gas density, vi is the gas velocity, and T is the gas temperature. The definitions of stress tensor and heat flux have already been given by Eqs. (2.41) and (2.42) in Sect. 2.1.3. The gas pressure p and the internal energy e are given by p = ρ RT and e = (3R/2)T , respectively.13 The collision term J ( f ) for the molecules with a spherically symmetric intermolecular potential with a finite influence range dm is given by a five-fold integral of f [35], relations p = ρ RT and e = (3R/2)T do not imply that the gas is in a (local) equilibrium state. They are formal extensions to nonequilibrium states, as well as the definition of temperature T .

13 The

40

2 Kinetic Boundary Condition at the Interface

J( f ) =

1 m

all αi , all ξi∗

( f f ∗ − f f ∗ ) B dΩ(α) dξ ∗ ,

(2.63)

where f = f (xi , ξi , t), f = f (xi , ξi , t), f ∗ = f (xi , ξi∗ , t), f ∗ = f (xi , ξi∗ , t), (2.64) = ξi∗ − αi α j (ξ j∗ − ξ j ), ξi = ξi + αi α j (ξ j∗ − ξ j ), ξi∗

B=B

α j (ξ j∗ − ξ j ) , |ξ ∗ − ξ | , |ξ ∗ − ξ |

(2.65)

(2.66)

and α is a unit vector expressing the variation of the direction of molecular velocity just before and after the collision, dΩ(α) is the solid-angle element in the direction of α, and the functional form of B is determined by the intermolecular force; for example, B = dm2 |α j (ξ j∗ − ξ j )|/2 for a gas consisting of hard-sphere molecules with diameter dm [35]. The mean free path of gas molecules, , is defined as the product of the molecular average speed ξ = (8RT /π )1/2 and the inverse of the mean collision frequency ν c [35], =

ξ , νc

(2.67)

where the mean collision frequency14 1 νc = ρm

f (ξ ) f (ξ ∗ ) B dΩ(α) dξ dξ ∗ ,

(2.68)

is an average of the collision frequency of a molecule with velocity ξ νc (ξ ) =

1 m

f (ξ ∗ ) B dΩ(α) dξ ∗ .

(2.69)

The integrations are taken over 6-dimensional space of (ξ , ξ ∗ ) in Eq. (2.68) and over 3-dimensional space of ξ ∗ in Eq. (2.69). When the velocity distribution function f is the Maxwellian (the Maxwell distribution function) with constant ρ0 , T0 , and v0i ,

f M (ξ ) =

14 The

ρ0 (ξi − v0i )2 , exp − 2RT0 (2π RT0 )3/2

inverse of the mean collision frequency is called the mean free time.

(2.70)

2.3

Kinetic Theory of Gases

41

the mean collision frequency ν c defined by Eq. (2.68) can be evaluated as ν c = 4dm2 (π RT0 )1/2

ρ0 , m

(2.71)

where ρ0 /m is the number density of molecules and, as mentioned earlier, dm is the radius of the influence range of the intermolecular force (not restricted to the diameter of a hard-sphere molecule) [35]. The mean free path is then given by 1 . = √ 2 2π dm (ρ0 /m)

(2.72)

Here, we should comment on the fundamental framework of the kinetic theory of gases based on the Boltzmann equation. The collision term of the Boltzmann equation only considers the binary collision of molecules. Mathematically, this is a situation where N → ∞ and dm → 0 with N dm2 fixed. This is called the Grad– Boltzmann limit [35]. In this limit, we have N dm3 → 0, and this means that the gas is an ideal gas. The N dm2 corresponds to the inverse of the mean free path of gas molecules,15 and it can be arbitrarily small or large as far as it is kept at a fixed value in the limiting process of N → ∞ and dm → 0. The wide applicability of the Boltzmann equation from atmospheric to very low pressures is therefore the consequence of the Grad–Boltzmann limit. In the right-hand side of Eq. (2.63), one can see a factor m −1 , which tends to infinity as m tends to zero. The collision term is, however, always bounded because the function B has a factor dm2 . In other words, it is implicitly assumed that the mass of a molecule m → 0 as N → ∞ with keeping N m fixed at a finite value, as it should be. This is the reason why we define the velocity distribution function by Eq. (2.59). In many literature, however, the velocity distribution function is defined by f /m, which is infinite in the Grad–Boltzmann limit. In addition to the hard-sphere model, the following model is widely used for the collision term of the Boltzmann equation (2.60): J ( f ) = Ac ρ( f e − f ), (ξ j − v j )2 ρ exp − fe = , 2RT (2π RT )3/2

(2.73) (2.74)

where Ac is a constant, f e is the local Maxwellian with density ρ, velocity v j , and temperature T defined by Eq. (2.62). The Boltzmann equation with the collision term (2.73) with Eq. (2.74) is called the Boltzmann–Krook–Welander (BKW) equation [35]. Since ρ, v j , and T in Eqs. (2.73) and (2.74) are given by the integrals of

15 Precisely, the mean free path corresponds to 1/(n d 2 ), where n is a characteristic number 0 m 0 density of gas molecules, as shown by Eq. (2.72).

42

2 Kinetic Boundary Condition at the Interface

unknown function f as shown in Eq. (2.62), the BKW equation is a highly nonlinear integro-differential equation. The constant Ac in Eq. (2.73) is related to the mean free path of molecules in the gas in an equilibrium state with density ρ and temperature T as =

(8RT /π )1/2 , Ac ρ

(2.75)

where Ac ρ is the collision frequency (νc ) of molecules described by the BKW equation. That is, the collision frequency of molecules described by the BKW equation is independent of the molecular velocity ξ [see Eqs. (2.68) and (2.69)], and hence νc = ν c . The BKW equation shares the important properties in the kinetic theory of gases with the standard Boltzmann equation with the collision term (2.63), (2.64), (2.65), and (2.66): (i) the Maxwellian (2.70) is the solution expressing the equilibrium state, where ρ0 , v0i , and T0 are constants, (ii) the same conservation equations for macroscopic variables can be derived as those from the standard Boltzmann equation, and (iii) the Boltzmann H-theorem16 holds [35]. By many theoretical and numerical studies, it has been confirmed that not only qualitatively but also quantitatively similar results are obtained for the BKW equation and the standard Boltzmann equation, except that the BKW equation gives the Prandtl number equal to unity [35]. Furthermore, the BKW equation has an advantage that two components of molecular velocity can be eliminated in spatially one-dimensional problems [35], and this considerably reduces the computational cost in simulation studies. The Gaussian– BGK Boltzmann equation [2] also has the same advantage, and as a result, we have obtained several substantial results in the analysis of shock-tube experiment for the condensation coefficients of water and methanol, as shown in Chap. 3.

16 The Boltzmann H-theorem corresponds to the entropy inequality extended to nonequilibrium states [35]. The theorem states that the H function or the integral H of H function over a domain D,

H=

f ln( f /c) dξ

or

H=

H dx, D

never increases by an inequality dH [1 + ln( f /c)]J ( f ) dξ dV 0, − (Hi − H vwi )n i dS = dt ∂D D if (Hi − H vwi )n i = 0 on the boundary ∂ D, where c is a constant to make f /c dimensionless and Hi = ξi f ln( f /c) dξ .

2.3

Kinetic Theory of Gases

43

2.3.2 Boundary Condition for the Boltzmann Equation The gas molecules impinging on the surface of solid or liquid are scattered by some rule. That is, the velocity distribution of molecules leaving the boundary should be regulated by some rule other than the molecular interaction law in the gas. This is the boundary condition for the Boltzmann equation and called the kinetic boundary condition. In the case of a solid boundary [see Fig. 2.6a], the commonly used one is the diffuse-reflection condition [35], f (x, ξ , t) =

% & [ξi − vwi (x, t)]2 ρw exp − , 2RTw (x, t) [2π RTw (x, t)]3/2

(2.76)

for molecules leaving the boundary with the velocity ξ satisfying ξ · n(x, t) > v w (x, t) · n(x, t),

(2.77)

at a point x on the surface of the boundary and at a time t, where Tw (x, t) and v w (x, t) are respectively the temperature and velocity at the point on the surface of the boundary, n(x, t) is the unit vector normal to the surface and pointing to the gas phase, and ρw = − ×

2π RTw (x, t)

1/2

ξ j n j (x,t)
( ' ξ j − vwj (x, t) n j (x, t) f (x, ξ , t) dξ .

(2.78)

The diffuse-reflection condition represents the situation that all the molecules impinging on the boundary are isotropically emitted from the boundary after the full adaptation to the velocity and temperature of the boundary in an infinitesimal time interval. From the diffuse-reflection condition (2.76), (2.77), and (2.78), it can immediately be confirmed that the mass flux across the boundary is equal to zero.

M=0 /

M=0 x

Gas Boundary

Solid

(a)

n

O

x

Vapor Interface

Liquid

(b)

n

O

Fig. 2.6 (a) The solid boundary and the unit normal vector pointing to the gas phase. (b) The vapor–liquid interface and the unit normal vector pointing to the vapor phase

44

2 Kinetic Boundary Condition at the Interface

In the case that the boundary is a vapor–liquid interface [see Fig. 2.6b], the simplest one is the complete-condensation condition [35], f (x, ξ , t) =

% & [ξi − vwi (x, t)]2 ρ∗ exp − , 2RTw (x, t) [2π RTw (x, t)]3/2

(2.79)

for molecules leaving the boundary with the velocity satisfying Eq. (2.77), where ρ ∗ is the saturated vapor density at temperature Tw (x, t). The mass flux M across the interface is then given by ∗

M(x, t) = (ρ − ρw )

RTw (x, t) , 2π

(2.80)

where Eq. (2.78) is used. The right-hand side of Eq. (2.79) is the same form as the Maxwell distribution in the vapor–liquid equilibrium state of uniform temperature Tw and uniform velocity v w . In the equilibrium state, the velocity distribution function of impinging molecules, appearing in the integrand of Eq. (2.78), is equal to the Maxwellian, and hence ρw defined by Eq. (2.78) is equal to ρ ∗ , which yields M = 0. The situation represented by the complete-condensation condition may be explained as follows: (i) all the molecules impinging on the interface are adsorbed to the interface; (ii) it is impossible or meaningless to distinguish which molecule leaving the boundary is reflected at the boundary or comes from the inside of the interface; (iii) the velocity distribution of molecules leaving the interface is independent of the vapor and solely determined by the interface properties, ρ ∗ , Tw , and v w . Both the diffuse-reflection and complete-condensation conditions have the same Maxwellian-like functional form with respect to the molecular velocity ξ . If we assume that the ξ -dependence of a boundary condition is unchanged irrespective of the gas condition, the Maxwellian-like functional form is essential because the boundary-value problem of the Boltzmann equation has an equilibrium solution of Maxwellian. Since the situations of diffuse-reflection and complete-condensation conditions are compatible and they have the same Maxwellian-like functional form with respect to ξ , another type of boundary condition can be considered [35]: % & [ξi − vwi (x, t)]2 αe ρ ∗ + (1 − αc )ρw exp − , f (x, ξ , t) = [2π RTw (x, t)]3/2 2RTw (x, t)

(2.81)

with Eq. (2.78) for molecules leaving the boundary with the velocity satisfying Eq. (2.77), where αe (0 αe 1) and αc (0 αc 1) are called the evaporation coefficient and condensation coefficient, respectively. This mixed-type boundary condition (2.81) has also widely been used so far, and is our target in this book.

2.4

Kinetic Boundary Condition

45

In the kinetic boundary condition (2.81), we assume that the information of molecules impinging on the interface is contained in (1 − αc )ρw only, and αe in the right-hand side of Eq. (2.81) is independent of molecules impinging on the interface. Under these assumptions, the αe part in Eq. (2.81), i.e., the complete-condensation condition multiplied by αe , is unchanged even if there are no molecules impinging on the interface. That is, the αe part does not include the molecules reflected at the interface, and hence αe is named the evaporation coefficient. The αe part in Eq. (2.81) is studied by a nonequilibrium MD simulation in Sect. 2.4.1. On the other hand, the (1 − αc ) part, i.e., the diffuse-reflection condition multiplied by (1 − αc ), may be regarded as the reflection part and the remaining αc fraction of ρw may be considered to be condensed on the interface. Hence, αc is called the condensation coefficient. The condensation coefficient αc and Eq. (2.81) itself are studied in Sect. 2.4.2 by a nonequilibrium MD simulation. Based upon the kinetic boundary condition (2.81), the mass flux across the interface is given by ∗

M = (αe ρ − αc ρw )

RTw (x, t) . 2π

(2.82)

This formula will be repeatedly utilized in this book. The formulation of the boundary-value problem for vapor adjacent to a vapor– liquid interface is completed by the Boltzmann equation (2.60), the boundary condition at the interface, Eqs. (2.81), (2.77), (2.78), and the boundary condition at infinity. We assume that the evaporation or condensation is sufficiently weak so that we can expect that the liquid is in a local equilibrium state. This allows us to determine the temperature field and velocity field in the liquid phase by the equations of fluid dynamics (see Appendix B at the end of this book). The temperature Tw (x, t) and velocity vwi (x, t) at the interface are thus determined without relying on the kinetic theory of liquids. Then, we can solve the boundary-value problem of the Boltzmann equation. Now, we have made all sorts of preparations for tackling our main problem.

2.4 Kinetic Boundary Condition The first topic of our main problem in this chapter is to establish the kinetic boundary condition at the vapor–liquid interface by the method of molecular dynamics (MD) simulation. In Sect. 2.3.2, we have introduced the boundary condition (2.81) as a widely used one. However, the derivation of (2.81) has not been accomplished neither from the kinetic theories of gases and liquids nor from the Liouville equation (2.16). Until the MD studies on (2.81) were performed [16–18], it has been just a mathematical model. In Sect. 2.4.1, according to Refs. [16–18], we firstly construct the αe part of Eq. (2.81) and determine the values of αe for argon, water, and methanol by specially devised nonequilibrium MD simulations. Then, in Sect. 2.4.2,

46

2 Kinetic Boundary Condition at the Interface

by using further devised MD simulation of nonequilibrium states, we formulate the kinetic boundary condition at the vapor–liquid interface, the result of which is substantially equal to Eq. (2.81) if the condensation mass flux across the interface is not so large.

2.4.1 Evaporation into Vacuum In Sect. 2.3.2, the αe part of Eq. (2.81) is assumed to be independent of the molecules impinging on the interface and unchanged even if no molecules impinge on the interface. We therefore consider a situation that all vapor molecules near the interface have the velocities in the direction of leaving the interface. This can be realized in a molecular simulation by eliminating the molecules impinging on the interface, and we call this type of simulation the MD simulation of evaporation from the interface into a virtual vacuum [16, 17]. Note that the vapor density near the interface cannot be zero in reality, and hence we call this situation the virtual vacuum. From the samples obtained by the simulation, we can directly construct the velocity distribution function of molecules evaporating from the interface, and thereby investigating the functional form of the velocity distribution function. Such an MD simulation of evaporation into vacuum has first been performed by Anisimov et al. [3]. The simulation is started from an equilibrium vapor–liquid two-phase system as shown in Fig. 2.3 [16, 17]. Although the equations solved are Newton’s equations of motion (2.47), (2.48), (2.49), and (2.50) in the nondimensional form, we present the results in the dimensional forms with the use of (/k, σ ) = (119.8 K, 0.341 nm) for argon. The leap-frog method with the time step 10−15 s and the cut-off radius 1.5 nm (4.4 σ ) are used. The evaporation into a virtual vacuum is actualized by eliminating vapor molecules in a region that is a few multiples of a thickness of interface distant from the center of the interface, where the center of the interface x 0 and the thickness of the interface δ are defined by

ρ L + ρV x0 = ρ , 2 ρ L − ρV ρ L − ρV −1 −1 ρV + ρL − δ=ρ −ρ , 10 10 −1

(2.83) (2.84)

where ρ L and ρV are the density of the liquid and vapor near the interface, and ρ −1 is the inverse function of ρ(x) (x is the space coordinate normal to the interface, which is a plane in the macroscopic sense). The thickness δ defined by Eq. (2.84) is called the 10–90 thickness (see Fig. 2.7). Since the evaporating molecules carry the energy from the liquid, the temperature of liquid drops with the time goes on. To maintain the temperature in the liquid, we apply the temperature control by the velocity scaling method [1], and thus the evaporation phenomenon observed on the coordinate fixed at the interface can be regarded as a steady state. The distance where the vapor molecules are eliminated and the region where the temperature

2.4

Kinetic Boundary Condition

47

ρL

ρV

ρL 10 ρL + ρ V 2 ρV

ρL

ρV

10

x

x0 δ

Fig. 2.7 The density profile (bold curve), the center of the interface x 0 , and the 10–90 thickness δ of the density transition layer

control is applied are carefully examined and chosen appropriately [16]. However, by the temperature control, the temperature in a bulk liquid region is rendered to be uniform. The simulation may therefore be an approximation for the case that the temperature gradient in the bulk liquid phase, which is necessarily generated by the heat flux due to evaporation, is sufficiently small. Figure 2.8 shows the averaged molecular mass fluxes in the simulation of evaporation into the virtual vacuum, where X = (x − x0 )/δ is a normalized space coordinate normal to the interface and measured from the center of the interface, and the molecular mass fluxes are averaged with respect to time on the coordinate fixed at the interface.17 The molecules are eliminated at X = 4. The liquid temperature controlled by the velocity scaling is shown in the figure. The angle brackets · · · in Fig. 2.8 represent a long-time average. The averaged mass flux M shown in the figure is defined by the time average of the molecular mass flux M

Molecular Mass Flux

(g/cm2.s) M+ (130K) M (130K)

103

M+ (85K)

102

M

101

0

1

2 X

(85K)

3

4

Fig. 2.8 The molecular mass fluxes in the simulation of evaporation into the virtual vacuum [16]. The liquid temperature controlled by the velocity scaling is shown in the figure. Here, the angle brackets represent a long-time average

17 The

center of the interface x0 is almost constant on the coordinate fixed to the interface in spite of the backward movement of interface due to evaporation.

48

2 Kinetic Boundary Condition at the Interface

M(x, t) =

N 1 χ 1 (q (i) − x, h) p(i) j n j F(q, p, t) dq d p, L 2 L3h

(2.85)

i=1

which is the one-dimensional counterpart of Eq. (2.20), χ 1 is defined by Eq. (2.58), and n j is the unit vector normal to the interface and pointing to the vapor phase. Another averaged mass flux M+ is defined by the time average of the molecular mass flux M+ 1 M+ (x, t) = L 2 L 3h

N

χ 1 (q (i) − x, h) p (i) j n j F(q, p, t) dq d p,

(2.86)

i=1 (i) p j n j >0

which gives the mass flux of molecules moving in the positive X direction. As shown in Fig. 2.8, in the region X 1, the region very close to the interface, the density is not low, and hence the molecular interactions frequently happen. The difference between M and M+ means that there exist some molecules moving in the negative X direction. In the region X 2, the difference of the two fluxes vanishes and all molecules are evaporating into the virtual vacuum. The velocity distribution function can be constructed according to its definition Eq. (2.59), and the results are shown in Fig. 2.9, where the normalized velocity distribution functions of the normalized molecular velocity (ζx , ζ y , ζz ) = √ (ξx , ξ y , ξz )/ 2RTL are plotted and the temperature of the liquid controlled by velocity scaling TL is also shown.18 The velocity component normal to the interface is signified by the open circle. At X = 0, as shown in Fig. 2.9a, d, the velocity distribution function is the Maxwellian with the temperature TL . Although X = 0 is not the bulk liquid phase but the center of the interface, the normalized velocity distribution of molecules is the same as that in the equilibrium state. However, at X = 2, the profiles of distribution of velocity component normal to the interface, ζx , are distorted as shown in Fig. 2.9c, e, which clearly means that the number of molecules with negative ζx decreases. In the case that X = 4 and TL = 85 K, the distribution of ζx > 0 is almost equal to the half part of the one-dimensional Maxwellian with the temperature TL and zero mean velocity, and the distribution of the other components are unchanged from those at X = 0. That is, the velocity distribution of molecules evaporating into the virtual vacuum has the same functional form as that of Eq. (2.81) at TL = 85 K. On the other hand, at TL = 130 K, a significant amount of molecules have the velocity of ζx < 0 even at X = 4. This means that, when the temperature of the liquid is high, the vapor density of molecules evaporating into the virtual vacuum is also high, and hence the molecular interaction in the vapor phase cannot be neglected.

18 In

Eq. (2.81), we used the symbol Tw for the temperature at the surface of the liquid phase.

2.4

Kinetic Boundary Condition

0.6

49

(a)

TL = 85K

ζx ζy ζz

Normalized Velocity Distribution Function

X= 0 0.3

0 –3 0.6

–2

–1

0

1

2

X= 0

3 –3

(b)

TL = 85K

(d)

TL = 130K

–2

–1

0

1

2

TL = 130K

X= 2

3

(e)

X= 2

0.3

0 –3 0.6

–2

–1

0

1

2

3 –3

(c)

TL = 85K X= 4

–2

–1

0

1

2

3

(f)

TL = 130K X= 4

0.3

0 –3

–2

–1

0 1 2 3 –3 –2 –1 0 Normalized Molecular Velocity

1

2

3

Fig. 2.9 The velocity distribution functions √ at X = 0, 2, 4 [16]. The abscissa of the figure is the normalized molecular velocity ζ j = ξ j / 2RTL ( j = x, y, z), where TL is the temperature of the liquid controlled by velocity scaling

In this case, it is impossible to realize the situation that all vapor molecules near the interface have the velocities in direction of leaving the interface. Note that the mean free path of the vapor molecules in the equilibrium at T = 130 K is about 2σ ≈ 2dm , and hence the vapor is not in the Grad–Boltzmann limit.

2.4.2 Evaporation Coefficient Thus, the velocity distribution function of molecules evaporating from the interface into the virtual vacuum is obtained by the MD simulation, and its functional form is found to be the half part of the Maxwellian with liquid temperature TL and zero mean velocity, if TL is not so high or if the vapor density is sufficiently low so that the gas may be regarded to be in the Grad–Boltzmann limit. That is, the functional form of the αe part of Eq. (2.81), which is independent of the molecules impinging on the interface, is validated by the MD simulation. Once admitted the αe part of Eq. (2.81), by considering the situation that ρw = 0 in Eq. (2.81), and applying Eq. (2.82), we have

50

2 Kinetic Boundary Condition at the Interface

Evaporation Coefficient αe

1 Argon 0.8 0.6 Ishiyama et al. (2004) Tsuruta et al. (2002) Anisimov et al. (1999) Matsumoto (1998)

0.4 0.2 0

70

80

90 100 110 Temperature (K)

120

130

Fig. 2.10 Evaporation coefficient for argon [16] and the comparison with those obtained by other authors [3, 25, 41]

M+ = αe ρ ∗

RTL 2π

1/2 .

(2.87)

The left-hand side of Eq. (2.87) has already been evaluated in the simulation as shown in Fig. 2.8, and the saturated vapor density ρ ∗ can be calculated from the Clausius–Clapeyron equation [30] or the equilibrium MD simulation as shown in Sect. 2.2.3. The values of the evaporation coefficient αe can thus be determined and the results are plotted in Fig. 2.10. In the figure, the evaporation coefficient for TL below the triple point temperature are also shown, where the sublimation occurs from the vapor–solid interface. Since the Lennard-Jones potential is not suitable for the vapor–solid equilibrium state, we have used the Dymond–Alder potential for the temperature range of sublimation [10, 16]. As can be seen from Fig. 2.10, the evaporation coefficient αe is a decreasing function of liquid temperature, and it is almost unity at slightly below the triple point temperature. We here remark that the left-hand side of Eq. (2.87), M+ , is evaluated by the MD simulation formulated in a nondimensional form, and hence the obtained M+ is applicable to other species of molecules, for which the LennardJones 12-6 potential is effective, e.g., Ne, Kr, and so on. The evaporation coefficient of other species of molecules can therefore be evaluated by using M+ obtained here and the saturated vapor density ρ ∗ of the species of molecules. The method of MD simulation of evaporation into the virtual vacuum has been extended to the cases of polyatomic molecules [17], for water and methanol. In the simulations [17], TIP3P model for water [20] and OPLS model for methanol [19] have been used for the intermolecular potentials, in addition to the cut and shifted Coulomb potentials [17]. For polyatomic molecules, the mathematical model of the kinetic boundary condition at the interface is given by [8]

2.4

Kinetic Boundary Condition

51

& % αe ρ ∗ + (1 − αc )ρw [ξi − vwi (x, t)]2 f (x, ξ , η, t) = exp − [2π RTw (x, t)]3/2 2RTw 1 η2/n , × exp − RTw (x, t) Γ (n/2 + 1)[RTw (x, t)]n/2

(2.88)

with Eq. (2.78) for molecules leaving the boundary with the velocity satisfying Eq. (2.77), where η2/n (0 η < ∞) is the energy of internal motion of one polyatomic molecule with the internal degrees of freedom n,19 and Γ is the gamma function. The Boltzmann equation and the kinetic boundary condition for polyatomic gas molecules is discussed in Sect. 2.5.1. As in the case of monatomic molecule [16], the αe part of the velocity distribution function of polyatomic molecules leaving the interface is found to be the half of the Maxwellian with temperature TL and zero mean velocity in the case that the liquid temperature is close to the triple point temperature [17]. Again, using Eq. (2.87), the evaporation coefficients for water and methanol are determined. The results are shown in Fig. 2.11. It may be worth noting that the temperature dependence of evaporation coefficients of polyatomic molecules is very similar to that of the monatomic molecule. 1 0.8

αe

0.6 0.4

Argon

} Water } Methanol

0.2 0 0.4

0.5

0.6

0.7 0.8 TL/Tcr

0.9

1

Fig. 2.11 Evaporation coefficients of argon, water, and methanol. The closed squares are from Ref. [16], the closed circles and solid triangles are from Ref. [17], the open circles are from Ref. [27], and the open triangles are from Ref. [25]. The abscissa of the figure is the liquid temperature TL divided by the critical temperature Tcr , where Tcr for argon, TIP3P model for water [20], and OPLS model for methanol [19] are 151, 516, and 404 K, respectively

19 The internal motions of a polyatomic molecule are the rotational and vibrational motions. Although the rotational motions are usually active at room temperature, the vibrational modes are activated at higher temperature. However, since n is constant in Eq. (2.88), the gas flows associated with the activation and deactivation of the vibrational modes cannot be treated by Eq. (2.88). The distribution of the energy of internal motion is discussed in Sect. 3.1.2.

52

2 Kinetic Boundary Condition at the Interface

2.4.3 Condensation Coefficient and KBC in Weak Condensation States In Sect. 2.4.2, we have constructed the velocity distribution function of molecules evaporating from the interface into the virtual vacuum, and confirmed that the αe parts of Eqs. (2.81) and (2.88) are precisely reproduced by the MD simulation for the case that the temperature of the liquid phase is low so that the vapor can be regarded as an ideal gas. Next, we construct the complete functional form of Eq. (2.81) by the nonequilibrium MD simulation of various steady condensation states of monatomic molecules [18]. The steady condensation states are realized in the MD simulation by controlling the velocity distribution of molecules impinging on the interface as ξi2 ρcol , exp − f = (2π RTcol )3/2 2RTcol

ρcol = βρ ∗ ,

Tcol = γ TL ,

(2.89)

where we studied 16 cases of β = 1, 2, 3, 4 and γ = 1, 2, 3, 4 [18]. Note that ρ ∗ is the saturated vapor density at the temperature TL , and the case of β = γ = 1 corresponds to the equilibrium state. The profiles of averaged density for (β, γ ) = (1, 1), (2, 2), and (4, 4) are shown in Fig. 1.1 in Chap. 1, where the parameters (a, b) are used in place of (β, γ ). The condensation at the interface means the transport of mass and energy across the interface, and hence the liquid temperature inevitably increases unless some amount of heat is absorbed appropriately inside the liquid phase. In the MD simulations in Ref. [18], we have fixed the temperature of bulk liquid phase TL by the velocity control method. This renders the temperature distribution in the bulk liquid phase spatially uniform in spite of the existence of heat flux. We therefore consider that our results are valid in the case that the condensation is weak and hence the temperature gradient in the bulk liquid is negligibly small. Figure 2.12 shows the velocity distribution of molecules leaving the interface at several pairs of (β, γ ), where the velocity distribution function is evaluated on the coordinate that moves with the interface. As clearly shown in the figure, the distribution functions of velocity component normal to the interface are almost the half of the one-dimensional Maxwellian with temperature TL and zero mean velocity [Fig. 2.12b–d]. On the other hand, the distribution functions of velocity component tangential to the interface deviate from the Maxwellian with temperature TL as the values of the pair (β, γ ) increase [Fig. 2.12e–h]. The results shown in Fig. 2.12 can be formulated into the kinetic boundary condition of the form 1 ξx2 f = [αe ρ + (1 − αc )ρw ] √ exp − 2RTL 2π RTL 2 ξy ξz2 1 1 ×√ , exp − exp − √ 2RTT 2RTT 2π RTT 2π RTT ∗

(2.90)

2.4

Kinetic Boundary Condition 0.6

Normalized Velocity Distribution Function

0.6

(β β =1, γ =1)

0.4 0.2

0 0.6

0 0.6

0 0.6

0.4

0

1 ζz

2

( β = 3, γ = 3)

(g)

0 0.6 0.4

(d)

0.2

TT = 136.6 K

0.2

(β = 4, γ = 4)

0.4

(f)

0 0.6

(c)

0.2

( β = 2, γ = 2)

0.2

(β = 3, γ = 3)

0.4

TT = 105.3 K

0.4

(b)

0.2

(e)

0.2

(β = 2, γ = 2)

0.4

(β =1, γ =1)

0.4

(a)

0 0.6

0

53

TT = 172.2 K

( β = 4, γ = 4)

(h)

0.2

3

0

–3

–2

–1

0 ζx Normalized Molecular Velocity

1

2

3

Fig. 2.12 Velocity distribution functions of molecules leaving the interface at several condensation states [18]. The temperature of the bulk liquid is fixed at TL = 85 K by the velocity control method. The equilibrium state at TL = 85 K and at rest is shown in the panels (a) and (e)

for the molecules leaving the interface with ξx > 0, where TT is the temperature associated with the velocity component tangential to the interface, as shown in Fig. 2.12f–h. In Ref. [18], we have shown that TT has a strong correlation with the energy flux across the interface, although a precise functional relation between TT and the energy flux still remains unresolved. Nevertheless, we can conclude that Eq. (2.81) can be retrieved for small (β, γ ), i.e., in the weak condensation states. Since we have obtained Eq. (2.90), we can determine the values of condensation coefficient αc at various nonequilibrium states. To do so, we rewrite the mass flux equation given by Eq. (2.82) as RTL 1/2 M = (αe ρ ∗ − αc ρw ) , (2.91) 2π by using Eq. (2.90), where the angle brackets indicate the time average. Substituting the evaporation coefficient αe obtained in Sect. 2.4.1, ρ ∗ , ρw given by Eq. (2.78), and the mass flux M obtained by the present MD simulations, we can evaluate the condensation coefficient αc from Eq. (2.91), and the results are plotted in Fig. 2.13. The figure clearly shows that αc is almost equal to αe = 0.868 at TL = 85 K [18]. We emphasize that if αc is a constant, it must be equal to αe , because f given by Eq. (2.90) should be equal to the Maxwellian with TL = TT and ρw = ρ ∗ at the equilibrium state.

2 Kinetic Boundary Condition at the Interface

αc

54

1 0.8 0.6 0.4 0.2 0 0

1

2

γ

3

4

5

0

1

2

3

4

5

β

Fig. 2.13 The condensation coefficient at several condensation states [18]. The open circle corresponds to the equilibrium state at temperature TL and at rest

2.5 Asymptotic Analysis of Weak Condensation State of Methanol In the preceding sections in this chapter, we have demonstrated that the relations connecting the liquid phase, the interface, and the vapor phase can be simplified and reformulated into the KBC at the interface, namely, Eq. (2.81) with Eqs. (2.77) and (2.78) for monatomic molecules and Eq. (2.88) with Eqs. (2.77) and (2.78) for polyatomic molecules. The prerequisites are that (i) the evaporation and condensation should be weak so that the heat flux may be small and the temperature in the liquid near the interface may be regarded as uniform, and that (ii) the temperature of the liquid is not high so that the density of the vapor near the interface may be sufficiently low. The second prerequisite is easily satisfied if the vapor near the interface can be approximated by an ideal gas, and this holds usually. Therefore, we concentrate on the first prerequisite, which can be satisfied if the mass flux across the interface induced by evaporation or condensation is not large. Since the KBC at the interface is specified, what we should do next is to solve the boundary-value problem of the Boltzmann equation, thereby deriving the boundary conditions for vapor flow in the fluid-dynamics region governed by the set of Navier–Stokes equations or the set of Euler equations. The method of the derivation of the boundary condition for the gas flow in the fluid-dynamics region was devised by Sone [34, 35] on the basis of the kinetic theory of gases. The steady (time-independent) problems associated with monatomic molecules have been solved thoroughly and completely by Sone and his colleagues (see, for example, Refs. [4, 33, 36], and references in his books [34, 35]), including the case of a mixture of a vapor and a noncondensable gas [37, 39].20 In the following, we apply Sone’s asymptotic theory to the case of polyatomic vapor [43]. 20

In Ref. [39], the ghost effect [35] induced by the noncondensable gas is found. The ghost effect has first been found in Ref. [38], and means a finite effect produced by an infinitesimal quantity. For example, in Ref. [38], Sone et al. discussed the temperature field in the limit Kn → 0 affected by the thermal creep flow that has already vanished in the limit Kn → 0.

2.5

Asymptotic Analysis of Weak Condensation State of Methanol

55

2.5.1 Problem and Formulation In the presence of evaporation or condensation at the vapor–liquid interface, the vapor near the interface cannot be in an equilibrium state. The typical length scale characterizing the nonequilibrium behavior of the vapor near the interface is the mean free path of the vapor molecules, . On the other hand, the characteristic length scale of variations in macroscopic variables in the fluid-dynamics region is in general different from , and let it be L, which is determined by a macroscopic characteristics of vapor flow, such as a linear dimension of a body in the flow. Then, the problem considered here is the case that the Knudsen number defined by Kn =

, L

(2.92)

is sufficiently small compared with unity. The nonequilibrium region near the interface is called the Knudsen layer [35]. As we will see later, under the condition that Kn 1, the vapor outside the Knudsen layer is in a local equilibrium state in the leading order of approximation, and hence the vapor flow can be determined by the macroscopic quantities. The appropriate equations governing the macroscopic quantities (the set of Navier–Stokes equations or the set of Euler equations21 ) are derived by applying the asymptotic theory for small Knudsen numbers [35]. The boundary conditions associated with the derived equations for the macroscopic quantities should also be derived from the asymptotic theory [35]. To do so, we have to solve the Boltzmann equation with the kinetic boundary condition at the vapor–liquid interface and the boundary condition at a distant region where the vapor is in a local equilibrium state. This is called the Knudsen layer analysis or the half-space problem of gas flow with evaporation or condensation [35], and extensively studied by Sone and his colleagues for the cases of monatomic molecules [4, 33–37, 39]. The Knudsen layer analysis will be explained in Sect. 2.5.3. In this subsection, we employ the Boltzmann equation of the Gaussian–BGK model22 for polyatomic molecules [2] ' ( ∂f p ∂f = + ξj G( f ) − f , ∂t ∂Xj μ(1 − ν + θ ν)

(2.93)

where f (X, ξ , η, t) is the distribution function of molecular velocity and internal motion for polyatomic vapor molecules, t is the time, X j is the jth component of

21 Actually, the equations governing the macroscopic quantities outside the Knudsen layer are not always equal to the set of Navier–Stokes equations and the set of Euler equations. Depending upon the macroscopic situation of the flow considered, the derived equations can contain some terms that are not included in the set of Navier–Stokes equations, and hence these equations are sometimes called the fluid-dynamics-type equations [35]. 22 In

Ref. [2], the Gaussian–BGK models for both cases of monatomic and polyatomic molecules are discussed. The Boltzmann H-theorem is proved for both cases.

56

2 Kinetic Boundary Condition at the Interface

the position vector X, ξ j is the jth component of the molecular velocity vector ξ , p is the pressure, μ is the viscosity, and θ and ν are nondimensional parameters (0 < θ 1 and − 12 ν < 1). The symbol G in the right-hand side of Eq. (2.93) represents a nonlinear functional of f given by 1 ρ exp − (ξi − vi )τi−1 (ξ − v ) G( f ) = ) j j j 2 (2π )3 det(τi j ) η2/n 1 , (2.94) exp − × Γ (n/2 + 1)(RTrel )n/2 RTrel where η2/n denotes the internal energy of one polyatomic molecule associated with the internal degrees of freedom n, Trel is a relaxation temperature defined in Eq. (2.98), τi j is a rectified stress tensor defined by Eq. (2.99), Γ is the gamma function, and det(τi j ) is the determinant of matrix τi j . The ratio of specific heats of polyatomic molecules γ is related to the parameter n by γ =

n+5 . n+3

(2.95)

The most important feature of the Gaussian–BGK Boltzmann equation (2.93) and (2.94) is that the mathematical proof of the Boltzmann H-theorem23 has been given for the parameters θ and ν in the range 0 < θ 1 and − 12 ν < 1 [2]. The H-theorem has not been proved for other models of Boltzmann equations for polyatomic molecules. The macroscopic variables are defined by the four-fold integrals of f with respect to ξ and η as follows: ρ=

ρvi =

f dξ dη,

ξi f dξ dη,

nρTint = 2

3ρ RTtr =

(ξ j − v j )2 f dξ dη, (2.96)

η

2/n

f dξ dη,

(3 + n)T = 3Ttr + nTint ,

ρΘi j =

(ξi − vi )(ξ j − v j ) f dξ dη,

(2.97)

p = ρ RT,

Trel = θ T + (1 − θ )Tint ,

(2.98)

ρτi j = θ pδi j + (1 − θ )[(1 − ν)ρ RTtr δi j + νρΘi j ],

(2.99)

where ρ is the density, vi is the vapor velocity, T is the temperature, Ttr and Tint are, respectively, the temperatures associated with the translational and internal motions of the molecule, δi j is the Kronecker delta, and ρΘi j is the stress tensor. The fourfold integration with respect to ξ and η is carried out over the whole space of ξ and 0 η < ∞. The parameters θ , ν, and n are chosen so that the Prandtl number 23 See

Footnote 16.

2.5

Asymptotic Analysis of Weak Condensation State of Methanol

57

and viscosity coefficients in the Gaussian–BGK model can be adapted for given (or experimental) values by the relation 1 2 Pr = < ∞, 3 1 − ν + θν

0

2n 1 − ν + θ ν n μb = , μ 3+n 3θ (3 + n)θ

(2.100)

(2.101)

for 0 < θ 1 and − 12 ν < 1, where Pr (= c p μ/κ) is the Prandtl number and μb is the bulk viscosity (c p is the specific heat at constant pressure and κ is the thermal conductivity coefficient). The case that θ = 1 and ν = n = 0 corresponds to the BKW equation for monatomic molecules introduced in Sect. 2.3.1, and methanol at room temperature may be modeled by (θ, ν, n) = (0.6471, −0.5, 6). The mean free path of the Gaussian–BGK model is given by √ μ 2 2RT . = (1 − ν + θ ν) √ p π

(2.102)

where p/[μ(1 − ν + θ ν)] is the mean collision frequency. As mentioned in Footnote 19 in this chapter, since the parameter n is a constant in the Gaussian–BGK model (2.93), (2.94), (2.95), (2.96), (2.97), (2.98), and (2.99), the activation and deactivation of vibrational modes of molecular internal motions cannot be described by Eqs. (2.93), (2.94), (2.95), (2.96), (2.97), (2.98), and (2.99). For the temperature range treated in this book, the molecular internal motions are the rotational motions. The boundary condition for the Gaussian–BGK Boltzmann equation (2.93) and (2.94) is given by [see Eq. (2.88)] % & [ξi − vwi (X, t)]2 αe ρ ∗ + (1 − αc )ρw exp − f (X, ξ , η, t) = 2RTw (X, t) [2π RTw (X, t)]3/2 η2/n 1 exp − × , (2.103) RTw (X, t) Γ (n/2 + 1)[RTw (X, t)]n/2 and ρw = − ×

2π RTw (X, t)

1/2

0η<∞ ξ j n j (X,t)
( ' ξ j − vwj (X, t) n j (X, t) f (X, ξ , η, t) dξ dη, (2.104)

58

2 Kinetic Boundary Condition at the Interface

for molecules leaving the boundary with the velocity satisfying ξ · n(X, t) > v w (X, t) · n(X, t),

(2.105)

at a point X on the interface and at a time t, where Tw (X, t) and v w (X, t) are the temperature and velocity at the interface, n(X, t) is the unit vector normal to the interface and pointing to the vapor phase. Equation (2.103) is an extension of the mixed-type boundary condition (2.81) to polyatomic molecules, and its functional form with respect to ξ and η is equal to the half of the equilibrium distribution function of polyatomic molecules with the ratio of specific heats γ = (n+5)/(n+3) at the equilibrium state with the temperature Tw and the mean velocity v w . The condition of a weak nonequilibrium state can be expressed as v √ 2RT 1, 0

p − p0 p 1, 0

T − T0 T 1, 0

(2.106)

where p0 and T0 are the pressure and temperature in a reference equilibrium state at rest. Equation (2.106) should be viewed as a sufficient condition for the weak evaporation/condensation state where the mass flux across the interface is so small that the liquid phase may be regarded as in a local equilibrium state and the boundary condition (2.103) may hold. In the asymptotic analysis for small Knudsen numbers, we have to specify how the quantities in Eq. (2.106) are small compared with the small Knudsen number Kn (see Sect. 2.5.2).

2.5.2 Asymptotic Analysis for Small Knudsen Numbers We study the time-independent solution for the boundary-value problem (2.93), (2.94), (2.95), (2.96), (2.97), (2.98), and (2.99) with (2.103), assuming that the condensation is weak in the sense of Eq. (2.106), and the Knudsen number defined by Eqs. (2.92) and (2.102) is sufficiently small compared with unity. According to Refs. [34, 35], we seek a moderately varying solution of Gaussian– BGK model for polyatomic gas in a power series for small Kn (S expansion [35]),

φ = kφ S1 + k 2 φ S2 + · · · ,

k=

√ π Kn 1, 2

(2.107)

where φ = ( f − f 0 )/ f 0 is a nondimensional distribution function, f 0 is an equilibrium distribution in a reference equilibrium state at rest, and Kn is the Knudsen number defined by Eqs. (2.92) and (2.102) at the reference state. The mean free path 0 is given by √ μ0 2 2RT0 , 0 = (1 − ν + θ ν) √ p0 π

(2.108)

2.5

Asymptotic Analysis of Weak Condensation State of Methanol

59

(μ0 is the viscosity in the reference state). The moderately varying solution is often called the fluid-dynamics part of the solution of the problem [35]. The application of the S expansion (2.107) means that we consider the situation where v √ 2RT

0

= O(k),

p − p0 p = O(k), 0

T − T0 T = O(k). 0

(2.109)

In general, the Boltzmann equation (2.60) can be nondimensionalized as, for the time-independent problems, ζj

∂φ 1 = Jˆ(φ), ∂x j k

(2.110)

√ where ζ j = ξ j / 2RT0 is the nondimensional molecular velocity, x j = X j /L is the nondimensionalized spatial coordinate, and k −1 Jˆ(φ) is the nondimensionalized collision term of Eq. (2.60). The Gaussian–BGK Boltzmann equation can also be expressed as Eq. (2.110). From Eq. (2.110), one can immediately see that Jˆ(φ) = 0 in the leading order of approximation, if the moderately varying solution φ and its derivative with respect to x j are of the order of k. That is, the moderately varying solution is a local equilibrium distribution function in the leading order of approximation, because the collision term of the Boltzmann equation vanishes when and only when the distribution function is an equilibrium or a local equilibrium distribution function. The moderately varying solution is, therefore, characterized by the macroscopic variables, i.e., the velocity, the density, and the temperature. In the following, we will derive the equations governing these macroscopic variables and the boundary conditions for the equations. Substituting the power series (2.107) into the Boltzmann equation (2.110), and equating the coefficients of the same powers of k, we have a series of equations ˆ S1 ) = 0, L(φ

(2.111)

ˆ S2 ) = Nˆ (φ S1 ) + ζ j ∂φ S1 , L(φ ∂x j

(2.112)

······ where Lˆ is a linearized integral operator and Nˆ is a nonlinear integral operator, both of which are derived from the nondimensional collision term Jˆ in Eq. (2.110), and their explicit forms depend on the collision term. The homogeneous linear integral equation for φ S1 , Eq. (2.111), has nontrivial solutions corresponding to a local equilibrium solution, and hence the inhomogeneous term of Eq. (2.112) has to satisfy adequate solvability conditions in order that φ S2 is determinable. In addition to the fact that the solvability conditions render the asymptotic expansion (2.107) uniformly valid in the region of xi = O(1), the solvability conditions themselves are the equations governing the macroscopic variables in the region of xi = O(1), i.e., in the fluid-dynamics region [35].

60

2 Kinetic Boundary Condition at the Interface

For the Gaussian–BGK Boltzmann equation, the procedures to derive the equations governing the macroscopic variables are summarized in Ref. [43] for the timeindependent problem and in Ref. [14] for the time-dependent (linear) problem. The results for the time-independent problem are summarized as

u j S1

∂ PS1 = 0, ∂ xi

(2.113)

∂u i S1 = 0, ∂ xi

(2.114)

∂u i S1 1 ∂ PS2 Pr ∂ 2 u i S1 =− + , ∂x j 2 ∂ xi 2 ∂ x 2j

(2.115)

∂τ S1 1 ∂ 2 τ S1 = , ∂x j 2 ∂ x 2j

(2.116)

u j S1

∂ω S1 ∂u i S2 + u i S1 = 0, ∂ xi ∂ xi ∂u i S2 ∂u i S1 1 ∂ + (u j S2 + ω S1 u j S1 ) =− u j S1 ∂x j ∂x j 2 ∂ xi

(2.117)

Pr ∂ 2 τ S1 PS3 + 2 ∂ x 2j

∂u j S1 Pr ∂ 2 u i S2 ∂u i S1 Pr ∂ + βτ S1 , + + 2 ∂ x 2j 2 ∂ xi ∂x j ∂ xi

u j S1

(2.118)

∂τ S2 ∂ PS2 ∂τ S1 2 u j S1 + (u j S2 + ω S1 u j S1 ) − ∂x j ∂x j 5+n ∂x j Pr = 5+n

∂u j S1 ∂u i S1 + ∂x j ∂ xi

2

1 ∂2 + 2 ∂ x 2j

β 2 τ S2 + τ S1 , 2

(2.119)

where the macroscopic variables ω Sm , u i Sm , τ Sm , and PSm (m = 1, 2, 3) are the expansion coefficients of√nondimensional density ω = (ρ − ρ0 )/ρ0 , nondimensional velocity u i = vi / 2RT0 , nondimensional temperature τ = (T − T0 )/T0 , and nondimensional pressure P = (1 + ω)(1 + τ ), respectively, and the nondimensional parameter β in Eqs. (2.118) and (2.119) comes from the assumption that μ = μ0 (1 + τ )β . We shall remark that (i) Eqs. (2.113), (2.114), (2.115), (2.116), (2.117), (2.118), and (2.119) are the same as those derived from the asymptotic analysis of the BKW equation [35], if n = 0, Pr = 1, and β = 1; (ii) the temperature associated with the molecular translational motion Ttr and that with the molecular internal motion Tint are equal to the vapor temperature T up to the order shown above; (iii) the bulk

2.5

Asymptotic Analysis of Weak Condensation State of Methanol

61

viscosity does not appear up to the order shown above, while it appears in the third order in the time-dependent problem [14]. From Eq. (2.113), the vapor pressure is spatially uniform in the leading order of approximation. Equation (2.114) is the solenoidal condition for the vapor velocity in the leading order. However, the vapor flow is a compressible flow. In fact, the incompressibility condition u i S1

∂ω S1 = 0, ∂ xi

(2.120)

does not hold due to Eq. (2.117), and the energy equation (2.116) is different from that of incompressible flows in fluid dynamics, 1 ∂ 2 τ S1 ∂τ S1 n+3 = . u i S1 n+5 ∂x j 2 ∂ x 2j

(2.121)

2.5.3 Boundary Condition for the Equations in Fluid-Dynamics Region The final topic of this chapter is the derivation of the boundary conditions for the macroscopic equations (2.113), (2.114), (2.115), and (2.116). Thereby, the microscopic information connecting the liquid phase and the vapor phase through the interface can be transformed into the relations in terms of macroscopic variables. The microscopic information is included in the macroscopic relations as numerical constants, called the slip coefficients [35]. The boundary conditions for the macroscopic equations and the slip coefficients are derived by solving the Boltzmann equation in the Knudsen layer with the kinetic boundary condition (2.103), (2.104), and (2.105), and the procedure is called the Knudsen layer analysis [35]. Since the characteristic length scale in the Knudsen layer is the mean free path of the vapor molecules, we introduce a stretched independent variable y normal to the interface, y=

(xi − xwi )n i , k

(2.122)

where xwi represents the coordinate of the point on the interface. The Boltzmann equation (2.110) can be expressed as24 24 Sects. 2.5.2 and 2.5.3 are concerned with the time-independent problem. However, even in a time-dependent problem, if the characteristic time scale t0 of the problem is large so that √ 0 /t0 2RT0 = O(k), then the time derivative term in the Boltzmann equation drops in the Knudsen layer in the leading order of approximation, and hence the vapor flow in the Knudsen layer can be treated as a time-independent flow in the leading order of approximation, where the time variable t is included as a parameter.

62

2 Kinetic Boundary Condition at the Interface

ζi n i

∂φ K 1 ˆ K 1 ), = L(φ ∂y

(2.123)

in the leading order of approximation, where the nondimensional distribution function φ is decomposed into the sum of the fluid-dynamics part φ S and the Knudsen layer correction φ K , and φ K is expanded as φ K = kφ K 1 + k 2 φ K 2 + · · · .

(2.124)

Furthermore, it is assumed that φ K decays faster than any inverse power of y [35]. Since the coordinate y is stretched, the fluid-dynamics region is located at y → ∞. The Knudsen layer analysis is therefore equivalent to the so-called half-space problem of the Boltzmann equation [35]. The important point is that the solution of the half-space problem of the Boltzmann equation exists only when the condition at infinity satisfies some relations [35]. Therefore, at the same time when we obtain the solution of the half-space problem or the solution in the Knudsen layer (Knudsen layer function), the condition for the macroscopic variables in the fluid-dynamics region (condition at infinity for the half-space problem) are also obtained, and the microscopic information is incorporated into the condition for the macroscopic variables. This is the boundary condition for the macroscopic equations in the fluid-dynamics region. If the kinetic boundary condition at the interface has a Maxwellian-like functional form with respect to ξ , the solution of the half-space problem can be described with some universal functions independent of the multiplication factor of the Maxwellian-like function [42]. In Ref. [42], although the proof has been given for the Boltzmann equation of monatomic molecules, its extension to the Gaussian– BGK Boltzmann equation is straightforward. For the Gaussian–BGK Boltzmann equation, the Knudsen layer analysis can be carried out in the same way as those for the BKW equation in Refs. [33, 36] (see also [35]), and thereby the Knudsen layer corrections and the boundary condition for the macroscopic equations with slip coefficients can be obtained. The results in the leading order of approximation are as follows: (u i S1 − u wi1 )ti = 0,

PS1 − Pw1

u i K 1 = 0, √ 1 − αc αe αe − αc ∗ = , C4 − 2 π u i S1 n i + αc αc αc

(2.125) (2.126) (2.127)

τ S1 − τw1 = d4∗ u i S1 n i , ω K 1 = u i S1 n i Ω4∗ (y),

(2.128) (2.129)

∗ τtrK 1 = u i S1 n i Θ4tr (y),

(2.130)

τintK 1 =

∗ u i S1 n i Θ4int (y).

(2.131)

2.5

Asymptotic Analysis of Weak Condensation State of Methanol

63

Here, (i) the fluid-dynamics parts, u i S1 , τ S1 , and PS1 , are evaluated on the interface,25 and they are independent of y. (ii) u wi1 , τw1 , and Pw1 are, respectively, the first expansion coefficients of the nondimensional velocity, temperature √ of the interface, and the saturated vapor pressure at the temperature Tw , e.g., vwi / 2RTw = ku wi1 + k 2 u wi2 + · · · , and so on. (iii) u wi n i should vanish in the time-independent boundary-value problem. (iv) u i K 1 , ω K 1 , τtrK 1 , and τintK 1 are, respectively, the Knudsen layer corrections for the velocity, density, and temperature associated with translational and internal motions. (v) n i and ti are unit vectors normal and tangential to the interface, respectively. (vi) Equation (2.127) is a generalization of the result from the complete-condensation condition [36] according to Ref. [42]. It is important to note that the condensation coefficient αc varies according with the flow condition, whereas the evaporation coefficient αe is constant if Tw is constant. In the limit to the vapor–liquid equilibrium state, the evaporation or condensation ceases, i.e., u i S1 n i → 0 in Eqs. (2.127) and (2.128), and the vapor temperature and pressure should be equal to Tw and p ∗ , respectively. Accordingly, αc → αe in the limit of u i S1 n i → 0. We therefore require another information that gives αc as a function of u i S1 n i . This is one of main topics of Sect. 3. A theoretical formulation for constructing αc as a linear function of u i S1 n i is shown in the next subsection. The values of slip coefficients C4∗ and d4∗ and the functional forms of Knudsen ∗ (y), and Θ ∗ (y) are dependent on the parameters θ and layer functions Ω4∗ (y), Θ4tr 4int ν in the Gaussian–BGK model and the internal degrees of freedom n. In the present study, we set θ = 0.6471, ν = −0.5, n = 6, Pr = 0.86, and the ratio of specific heats 1.22 corresponding to methanol vapor at room temperature. In the case, we obtain [43]26 C4∗ = −2.0723,

d4∗ = −0.2185,

(2.132)

and Ω4∗ (0) = 0.475,

∗ Θ4tr (0) = −0.083,

∗ Θ4int (0) = 0.145.

(2.133)

The slip coefficients and the Knudsen layer functions for the BKW model and for the Boltzmann equation for hard-sphere gas are precisely determined and tabulated in the book [35]: for hard-sphere gas, C4∗ = −2.1412, Ω4∗ (0) = 0.37815,

d4∗ = −0.4557, Θ4∗ (0) = 0.05206,

(2.134) (2.135)

25 The fluid-dynamics parts are unchanged in the Knudsen layer and the Knudsen-layer corrections

rapidly vanish y → ∞. Therefore, u i S1 , τ S1 , and PS1 in Eq. (2.125) and Eqs. (2.127), (2.128), (2.129), (2.130), and (2.131) are, respectively, equal to the velocity, temperature, and pressure of the vapor at the outer edge of the Knudsen layer in the approximation of O(k). 26 These values have recently been corrected by M. Inaba as follows (private communication): C4∗ = −2.0719, d4∗ = −0.1921, ∗4 (0) = 0.5006, ∗4tr (0) = −0.1090, and ∗4int (0) = 0.1190.

64

2 Kinetic Boundary Condition at the Interface

and for the BKW model, C4∗ = −2.13204, Ω4∗ (0) = 0.36303,

d4∗ = −0.44675,

Θ4∗ (0)

= 0.03717.

(2.136) (2.137)

As can be seen from Eqs. (2.132), (2.134), and (2.136), C4∗ for the Gaussian– BGK model for polyatomic molecules is rather close to those for monatomic gas. The coefficient d4∗ for polyatomic molecules is, however, about a half of that for monatomic gas for the present case of θ = 0.6471, ν = −0.5, and n = 6. The dimensional expressions of Eqs. (2.127) and (2.128), extended to timedependent problems,27 can be written as p − p∗ αe = ∗ p αc

√ 1 − αc [vi − vwi (X, t)] n i (X, t) αe − αc ∗ + C4 − 2 π √ , αc αc 2RTw (X, t) (2.138) [v − v (X, t)] n (X, t) T − Tw (X, t) i wi i , (2.139) = d4∗ √ Tw (X, t) 2RTw (X, t)

where the saturated pressure p∗ = ρ ∗ RTw is a function of X and t through Tw (X, t).

2.5.4 Condensation Coefficient as a Linear Function of Mass Flux In the weak nonequilibrium problems in the sense of Eq. (2.109), all the perturbations, including the difference between αc and αe , are of the order of k. We may therefore assume that28 αc = αe + Λ

(vi − vwi )n i √ , 2RTw

(2.140)

where Λ is independent of (vi − vwi )n i and may be a function of Tw . The factor Λ contains the microscopic information of interface, which cannot be deduced by the kinetic theory of gases. The determination of the factor Λ may be possible by detailed nonequilibrium MD simulations, although it has not been performed yet. The experimental study presented in Sect. 3 is an only successful achievement for the determination of Λ up to now. By using Eq. (2.140), Eq. (2.138) is transformed as √ √ p 2 π Λ (vi − vwi )n i ∗ = 1 + C4 + 2 π − − , (2.141) √ p∗ αe αe 2RTw

27 See

Footnote 24.

√ dimensionless mass flux ρ(vi − vwi )n i /(ρ ∗ 2RT √ √w ) is approximately equal to (vi − vwi )n i / 2RTw in the accuracy of O(k) since (vi −vwi )n i / 2RTw = O(k) and ρ/ρ ∗ = 1+O(k).

28 The

2.5

Asymptotic Analysis of Weak Condensation State of Methanol

65

in the approximation up to O(k). The boundary conditions for the fluid-dynamics region are Eqs. (2.141) and (2.139). In Ref. [21], a nondimensional parameter A is defined by A=

αe ρ ∗ − αc ρw , ρ ∗ − ρw

(2.142)

and evaluated as constants, A = 0.52 for water and A = 0.50 for methanol, from the analysis of a number of data obtained by the shock-tube experiment of weak condensation at room temperature (see Sect. 3).29 We here derive the relation between the factor Λ in Eq. (2.140) and the parameter A. In the mass flux equation, RTw ∗ , (2.143) M = (αe ρ − αc ρw ) 2π M denotes the mass flux across the interface. Since the vapor flow in the Knudsen layer concerned is a time-independent flow, M can be replaced by ρ(vi − vwi )n i at the outer edge of the Knudsen layer, and we have ∗

ρ(vi − vwi )n i = (αe ρ − αc ρw )

RTw . 2π

(2.144)

Dividing Eq. (2.138) by Eq. (2.139) and neglecting the terms of O(k 2 ) gives √ √ αe 2 π (vi − vwi )n i ρ ∗ ∗ 1 + C , = − d + 2 π − √ 4 4 ρ∗ αc αc 2RTw

(2.145)

where αc has been treated as O(1) and Eq. (2.140) has not been used yet. Substituting ρ/ρ ∗ = αe /αc obtained from Eq. (2.145) into Eq. (2.144) yields √ 2 π (vi − vwi )n i αe ρw 1 − , = √ ρ∗ αc αc 2RTw

(2.146)

in the approximation up to O(k), where we still do not use Eq. (2.140). Substituting Eqs. (2.140) and (2.146) into Eq. (2.142), and taking the limit of (vi − vwi )n i → 0, we obtain " √ ! αe Λ=2 π −1 . (2.147) A Using αe = 0.86 for both water and methanol [17], we can determine Λ as follows: Λ = 2.3 for water, 29 Apparently,

Λ = 2.6 for methanol.

(2.148)

the right-hand side of Eq. (2.142) is equal to the ratio of the mass flux given by Eq. (2.82) to that in the case of complete-condensation condition given by Eq. (2.80).

66

2 Kinetic Boundary Condition at the Interface

2.6 Criticism on Hertz–Knudsen–Langmuir and Schrage Formulas In Sect. 2.5.3, the boundary conditions for the macroscopic equations in the fluiddynamics region, Eqs. (2.127) and (2.128), are derived by solving the Boltzmann equation in the Knudsen layer on the interface. Without solving the Boltzmann equation, the boundary conditions can never be obtained. However, several formulas have been proposed without solving the Boltzmann equation, and surprisingly, these formulas are used as the boundary conditions for the macroscopic equations in various applications still now. They are constructed from the mass flux equation (2.82) by replacing the unknown variable ρw with some functions of ρ, T , and (vi −vwi )n i at the outer edge of the Knudsen layer [7]. In the following, we take up typical two formulas and compare them with Eqs. (2.127) and (2.128). The first one is known as the Hertz–Knudsen–Langmuir formula [7], M HKL = √

1

2π R

p∗ p − αc √ αe √ Tw T

,

(2.149)

where αe and αc are the evaporation and condensation coefficients, for which αe = αc = α is sometimes used. The second one is the Schrage formula [7], M Sch =

2α √ (2 − α) 2π R

p p∗ −√ √ Tw T

,

(2.150)

where α is a nondimensional parameter for evaporation and condensation. Both Eqs. (2.149) and (2.150) are produced by replacing ρw in Eq. (2.82) by some functions consisted of p, T , and (vi − vwi )n i , although there is no justification for such a replacement without solving the Boltzmann equation. The inappropriateness of the Schrage formula can easily be demonstrated: In fact, for given Tw and p∗ , M Sch vanishes if a pair ( p, T ) satisfies the relation T = Tw

p p∗

2 .

(2.151)

That is, the mass flux across the interface vanishes for an infinite number of pairs ( p, T ) that satisfy Eq. (2.151) in spite of vapor–liquid nonequilibrium states. This contradicts the fact that the mass flux evaluated from the solution of the Boltzmann equation vanishes only in the vapor–liquid equilibrium state, i.e., p = p∗ and T = Tw , as shown by Eqs. (2.127) and (2.128). For the Hertz–Knudsen–Langmuir formula, the same discussion can be applied for pairs ( p, T ) satisfying T = Tw instead of Eq. (2.151).

αc p αe p ∗

2 ,

(2.152)

References

67

Furthermore, we express a pair ( p, T ) satisfying Eq. (2.151) or (2.152) as ( p † , T † ), and consider the linearization around ( p† , T † ). Substituting p = p† × (1 + P † ) and T = T † (1 + τ † ) into Eqs. (2.149) and (2.150), and linearizing the resulting equations under the assumption of |P † | 1 and |τ † | 1, we have 1 † αe p ∗ τ − P† , M HKL = √ 2π RTw 2 1 † 2α p∗ M Sch = √ τ − P† . 2 − α 2π RTw 2

(2.153) (2.154)

That is, the both formulas predict that the evaporation (M > 0) occurs when τ † > 2P † even if T − Tw > 0, and the condensation (M < 0) occurs when τ † < 2P † even if p − p∗ < 0. However, as can be seen from Eqs. (2.127) and (2.128) and from Fig. 2.14, the evaporation (u i S1 n i > 0) occurs only when T − Tw < 0 and the condensation (u i S1 n i < 0) occurs only when p − p ∗ > 0 (if αe = αc = 1). Although Eqs. (2.127) and (2.128) are results in the leading order of approximation for u i S1 n i = O(k), according to the numerical solution for nonlinear problems [35, 42], it is confirmed that the steady evaporation occurs only when T − Tw < 0 and the steady condensation occurs only when p − p∗ > 0. Thus, the both formulas are wrong and of no use. T

(p*,:Tw w) Evaporation

:

Tw w

M>0

Condensation M<0

(p†*,:Tw†) τ †..> .2P † M >0

τ.†. <. 2P † M<0

p p* Fig. 2.14 The boundary conditions for macroscopic equations, Eqs. (2.127) and (2.128), are shown by a bold straight line passing through the vapor–liquid equilibrium point ( p ∗ , Tw ) in the ( p, T ) plane. The parabola in the figure means Eqs. (2.151) or (2.152). The Hertz–Knudsen–Langmuir and Schrage formulas divide the neighborhood of a zero-mass-flux point ( p† , T † ) on the parabola into the evaporation and condensation regions with a thin dashed line τ † = 2P †

References 1. M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids (Clarendon Press, Oxford, 1987) 2. P. Andries, P.L. Tallec, J.P. Perlat, B. Perthame, Gaussian – BGK model of Boltzmann equation with small Prandtl number. Eur. J. Mech. B-Fluids 19, 813–830 (2000)

68

2 Kinetic Boundary Condition at the Interface

3. S.I. Anisimov, D.O. Dunikov, S.P. Malyshenko, V.V. Zhakhovskii, Properties of a liquid – gas interface at high-rate evaporation. J. Chem. Phys. 110, 8722–8729 (1999) 4. K. Aoki, K. Nishino, Y. Sone, H. Sugimoto, Numerical analysis of steady flows of a gas condensing on or evaporating from its plane condensed phase on the basis of kinetic theory: Effect of gas motion along the condensed phase. Phys. Fluids A3, 2260–2275 (1991) 5. M. Born, H.S. Green, A kinetic theory of liquids. Nature 4034, 251–254 (1947) 6. D.W. Brenner, Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films. Phys. Rev. B 42, 9458–9471 (1990) 7. H.K. Cammenga, in Current Topics in Materials Science, vol. 5, ed. by E. Kaldis. (NorthHolland, Amsterdam, 1980), pp. 335–446 8. C. Cercignani, Rarefied Gas Dynamics (Cambridge University Press, New York, NY, 2000) 9. S. Chapman, T.G. Cowling, The Mathematical Theory of Non-uniform Gases, 3rd edn. (Cambridge University Press, Cambridge, 1990) 10. J.H. Dymond, B.J. Alder, Pair potential for argon. J. Chem. Phys. 51, 309–320 (1969) 11. J.M. Hale, Molecular Dynamics Simulation, Elementary Methods (Wiley, New York, NY, 1992) 12. C.D. Holcomb, P. Clancy, S.M. Thompson, J.A. Zollweg, A critical study of simulations of the Lennard-Jones liquid – vapor interface. Fluid Phase Equilib. 75, 185–196 (1992) 13. W.G. Hoover, Molecular Dynamics. Lecture notes in physics vol. 258, (Springer, Berlin, 1986) 14. M. Inaba, S. Fujikawa, T. Yano, Molecular gas dynamics on condensation and evaporation of water induced by sound waves. Rarefied Gas Dynamics. AIP Conference Proceedings, vol. 1084 (AIP, Melville, NY, 2009), pp. 671–676 15. J.H. Irving, J.G. Kirkwood, The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics. J. Chem. Phys. 18, 817–829 (1950) 16. T. Ishiyama, T. Yano, S. Fujikawa, Molecular dynamics study of kinetic boundary condition at an interface between argon vapor and its condensed phase. Phys. Fluids 16, 2899–2906 (2004) 17. T. Ishiyama, T. Yano, S. Fujikawa, Molecular dynamics study of kinetic boundary condition at an interface between a polyatomic vapor and its condensed phase. Phys. Fluids 16, 4713–4726 (2004) 18. T. Ishiyama, T. Yano, S. Fujikawa, Kinetic boundary condition at a vapor – liquid interface. Phys. Rev. Lett. 95, 084504 (2005) 19. W.L. Jorgensen, Optimized intermolecular potential functions for liquid alcohols. J. Phys. Chem. 90, 1276–1284 (1986) 20. W.L. Jorgensen, J. Chandrasekhar, J.D. Madura, R.W. Impey, M.L. Klein, Comparison of simple potential functions for simulating liquid water. J. Phys. Chem. 79, 926–935 (1983) 21. K. Kobayashi, S. Watanabe, D. Yamano, T. Yano, S. Fujikawa, Condensation coefficient of water in a weak condensation state. Fluid Dyn. Res. 40, 585–596 (2008) 22. L.D. Landau, E.M. Lifshitz, Statistical Physics, 3rd edn. Part 1, Volume 5 of Course of Theoretical Physics (Elsevier, Amsterdam, 1980) 23. J.E. Lennard-Jones, The equation of state of gases and critical phenomena. Physica 4, 941–956 (1937) 24. E.M. Lifshitz, L. Pitaevskii, Physical Kinetics. Volume 10 of Course of Theoretical Physics (Elsevier, Amsterdam, 1980) 25. M. Matsumoto, Molecular dynamics of fluid phase change. Fluid Phase Equilib. 144, 307–314 (1998) 26. M. Mecke, J. Winkelmann, J. Fisher, Molecular dynamics simulation of the liquid-vapor interface: The Lennard-Jones fluid. J. Chem. Phys. 107, 9264–9270 (1997) 27. G. Nagayama, T. Tsuruta, A general expression for the condensation coefficient based on transition state theory and molecular dynamics simulation. J. Chem. Phys. 118, 1392–1399 (2003) 28. NIST Chemistry WebBook, http://webbook.nist.gov/chemistry/

References

69

29. W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes, 3rd edn. The Art of Scientific Computing (Cambridge University Press, Cambridge, 2007) 30. F. Reif, Fundamentals of Statistical and Thermal Physics (Waveland Press, Long Grove, IL, 2009) 31. P. Résiboirs, M. De Leener, Classical Kinetic Theory of Fluids (Wiley New York, NY, 1977) 32. A. Sommerfeld, Thermodynamics and Statistical Mechanics. Lectures on theoretical physics, vol V (Academic, New York, NY, 1964) 33. Y. Sone, Kinetic theory of evaporation and condensation – linear and nonlinear problems. J. Phys. Soc. Jpn. 45, 315–320 (1978) 34. Y. Sone, Kinetic Theory and Fluid Dynamics (Birkhäuser, Boston, MA, 2002) 35. Y. Sone, Molecular Gas Dynamics (Birkhäuser, Boston, MA, 2007) 36. Y. Sone, Y. Onishi, Kinetic theory of evaporation and condensation – hydrodynamic equation and slip boundary condition. J. Phys. Soc. Jpn. 44, 1981–1994 (1978) 37. Y. Sone, K. Aoki, T. Doi, Kinetic theory analysis of gas flows condensing on a plane condensed phase: Case of a mixture of a vapor and a noncondensable gas. Transp. Theory Stat. Phys. 21, 297–328 (1992) 38. Y. Sone, K. Aoki, S. Takata, H. Sugimoto, A.V. Bobylev, Inappropriateness of the heatconduction equation for description of a temperature field of a stationary gas in the continuum limit: Examination by asymptotic analysis and numerical computation of the Boltzmann equation. Phys. Fluids 8, 628–638 (1996); Erratum, ibid. 8, p. 841. 39. S. Takata, K. Aoki, Two-surface problems of a multicomponent mixture of vapors and noncondensable gases in the continuum limit in the light of kinetic theory. Phys. Fluids, 11, 2743–2756 (1999) 40. A. Trokhymchuk, J. Alejandre, Computer simulations of liquid/vapor interface in LennardJones fluids: Some questions and answers. J. Chem. Phys. 111, 8510–8523 (1999) 41. T. Tsuruta, G. Nagayama, Molecular dynamics study on condensation coefficients of water. Trans. Jpn. Soc. Mech. Eng. 68, 1898–1904 (2002) (in Japanese) 42. T. Yano, Half-space problem for gas flows with evaporation or condensation on a planar interface with a general boundary condition. Fluid Dyn. Res. 40, 474–484 (2008) 43. T. Yano, K. Kobayashi, S. Fujikawa, Condensation of methanol vapor onto its liquid film on a solid wall behind a reflected shock wave. Rarefied Gas Dynamics. AIP Conference Proceedings, vol. 762 (AIP, Melville, NY, 2005), pp. 208–213

Chapter 3

Methods for the Measurement of Evaporation and Condensation Coefficients

Abstract Based on the condensation characteristics made clear in Chapters 1 and 2, the weak condensation in nonequilibrium states of water and methanol vapors is realized in the vapor region between the shock tube endwall covered with thin liquid film for each vapor and the shock wave reflected there. The sophisticated method for the optical measurement of the growth speed of liquid film is proposed, examined in detail for prerequisites in the measurement, and clarified to be valid. Experimental data of the growth speed acquired are processed together with numerical solutions of the Gaussian–BGK Boltzmann equation applicable to polyatomic gases. The values of the condensation coefficient αc of both vapors are found to decrease from the respective αe -values, i.e., 0.99 for water and 0.86 for methanol as the interfacial vapor pressures increase under nearly constant liquid temperatures at about 290 K. It is also shown that the αe -values are determined by the liquid temperatures and material properties, while the αc -values are strongly dependent on the interfacial vapor states. A new method for the measurement of αe is also proposed on the basis of a theory of molecular gas dynamics, by which αe is expressed as a function of the amplitude of standing sound waves between a planar sound source and a vapor/liquid interface facing against it.

3.1 Review of αe , αc, KBC, and Gaussian–BGK Boltzmann Equation 3.1.1 Definitions of αe and αc The method of measurement of condensation coefficient consists of the shock-tube experiment and the numerical analysis of the Gaussian–BGK Boltzmann equation. In Sect. 3.1, we provide the basic concepts of evaporation and condensation coefficients and mathematical tools for the numerical analysis. As mentioned in Sect. 2.5, the KBC at the vapor–liquid interface for the weak evaporation or condensation of a polyatomic vapor is given by Eq. (2.103). In this section, we shall consider once again the KBC, the evaporation coefficient αe and

S. Fujikawa et al., Vapor-Liquid Interfaces, Bubbles and Droplets, Heat and Mass C Springer-Verlag Berlin Heidelberg 2011 Transfer, DOI 10.1007/978-3-642-18038-5_3,

71

72

3 Methods for the Measurement of Evaporation and Condensation Coefficients

Jevap

Jref

Jcoll

Bulk Vapor Phase Interface Transition Layer

Bulk Liquid Phase

Jcnds

Fig. 3.1 Definitions of molecular fluxes across the interface. The density changes continuously from the bulk liquid to vapor phase facing each other with the transition layer between

condensation coefficient αc contained in the KBC in more detail from physical aspects, because the sound understanding of their meanings is of importance for applications of the KBC and interpretation of αe and αc measured. The definitions of αe and αc are respectively given by molecular mass fluxes at the interface as follows [13–15]: αe =

ρ∗

√

Jevap , RTw /(2π )

αc =

Jcnds , Jcoll

(3.1)

√ where ρ ∗ RTw /(2π ) is a one-way molecular mass flux in the equilibrium state at the temperature of liquid Tw at the interface, ρ ∗ is the saturated vapor density at Tw , R is the gas constant per unit mass, Jevap is the mass flux of spontaneously evaporating liquid molecules across the interface, Jcoll is the mass flux of vapor molecules colliding against the interface, Jcnds (= Jcoll − Jref ) is the condensation mass flux, and Jref is the reflection mass flux. The schematic diagram of these molecular fluxes is depicted in Fig. 3.1. By the definition of Jevap , αe is independent of the collision of vapor molecules against the interface and it is dependent on Tw and physical properties of the material, i.e., R and ρ ∗ ; Jevap is governed by the collision of liquid molecules near the vapor-side edge of the transition layer. On the other hand, αc is dependent on the collision of both vapor and liquid molecules at the interface. It should be noted that αe and αc both take values between zero and unity according to the definitions; 0 αe 1 and 0 αc 1.

3.1.2 Extension of Monatomic Version of KBC to Polyatomic One We shall start our discussion from the KBC for monatomic molecules which has been formulated in Ref. [15] and is explained in Sects. 2.4 and 2.5 of Chap. 2. Figure 3.2 shows the coordinate system of molecular velocity, ξ = (ξx , ξ y , ξz ), in the vapor facing the interface moving with the velocity vw (> 0); the arrows indicate

3.1

Review of αe , αc , KBC, and Gaussian–BGK Boltzmann Equation

73

Fig. 3.2 The coordinate system of molecular velocity (ξx , ξ y , ξz ) in the vapor facing the interface moving with the speed vw toward the vapor

the positive directions of velocity components. The velocity distribution function of molecules leaving the interface into the vapor phase is given by1 f out (X = 0, ξ , t) = [αe ρ ∗ + (1 − αc )σ ] fˆtr ,

(ξx > vw ),

(3.2)

where (ξx − vw )2 + ξ y2 + ξz2 1 exp − , fˆtr = 2RTw (2π RTw )3/2 σ =−

2π RTw

(3.3)

ξx
(ξx − vw ) f coll dξ ,

(3.4)

where X is the space coordinate stemming from the interface into the vapor, f coll (X = 0, ξ , t) is the velocity distribution function for monatomic molecules colliding against the interface. Equation (3.2) means that the fraction (1 − αc ) of the vapor molecules colliding against the interface is reflected there and the remaining molecules condense, and on the other hand, the fraction αe of the molecules having the equilibrium velocity distribution with the saturated vapor density ρ ∗ at Tw spontaneously evaporates independently of the colliding vapor molecules. Equation (3.2) can be extended to the version for polyatomic molecules in the form consistent with the Gaussian–BGK Boltzmann equation [1, 18]. Considering that the velocity distribution function f out for outgoing monatomic molecules is given by the product of the Gauss distribution function (3.3) and the factor

1

Equation (3.2) is the spatially one-dimensional version of Eq. (2.103) for the case where molecular internal motions are excluded.

74

3 Methods for the Measurement of Evaporation and Condensation Coefficients

including αe and αc , as Eq. (3.2), we may express the distribution function of molecular velocity and internal motion for polyatomic molecules at the interface by the product of f out for the monatomic molecules and a distribution function for energy associated with the internal motions of a polyatomic molecule. We shall assume that the equipartition of energy for the internal motions holds in the equilibrium state at the temperature Tw . The partition function z for n modes on the internal motions is given by [30] z=

∞

p 2 + p22 + · · · pn2 exp − 1 ··· RTw −∞

d p1 d p2 · · · d pn ,

(3.5)

where p1 , p2 , . . . , pn are momentum components for the internal modes of the molecule and the number n is the internal degrees of freedom of the molecule. Note that n is related to the internal modes only, and it does not include the translational mode. Integrating Eq. (3.5) with respect to each variable, we obtain z = (π RTw )n/2 .

(3.6)

On the other hand, let us consider a sphere with radius r in the n-dimensional space ( p1 , p2 , . . . , pn ), whose origin coincides with the center of the sphere, with respect to the variables p1 , p2 , . . . , pn . With the use of the relation r 2 = p12 + p22 + · · · + pn2 and the sphere volume an r n in the n-dimensional space, we can express Eq. (3.5) as

∞

z = nan 0

r2 r n−1 dr. exp − RTw

(3.7)

Putting r 2 = η2/n , we can rewrite Eq. (3.7) as z = an 0

∞

η2/n dη, exp − RTw

(3.8)

where the coefficient an can be determined by putting Eq. (3.8) to be equal to Eq. (3.6). We can regard η2/n in Eq. (3.8) as the whole energy per unit mass for the internal modes of one molecule, and therefore we can define the distribution function for them as exp(−η2/n /RTw ) , fˆint = ∞ exp(−η2/n /RTw )dη

(3.9)

0

because the distribution function represents the probability density with which the internal modes of the molecule with energy η2/n appear. The variable η is called

3.1

Review of αe , αc , KBC, and Gaussian–BGK Boltzmann Equation

75

the internal parameter. The denominator of the right-hand side in Eq. (3.9) can be rewritten as2 ∞ exp(−η2/n /RTw )dη = Γ (n/2 + 1) (RTw )n/2 . (3.10) 0

With the use of Eq. (3.10), we can express Eq. (3.9) as fˆint =

1 η2/n . exp − RTw Γ (n/2 + 1)(RTw )n/2

(3.11)

Here, the value of n and the ratio of specific heats γ (= c p /cv ) are related by γ =

n+5 . n+3

(3.12)

For example, for water vapor at room temperatures, γ = 4/3 because n = 3 [30]. The distribution function at the interface can be constructed by the product of Eqs. (3.2) and (3.11) as f out (X = 0, ξ , η, t) = [αe ρ ∗ + (1 − αc )σ ] fˆtr fˆint ,

(ξx > vw ),

(3.13)

where ⎫ ⎪ ⎪ ⎪ ⎪ σ =− (ξx − vw ) f dξ dη, ⎪ ⎪ ⎪ ξx
2π RTw

coll

(3.14)

where dξ = dξx dξ y dξz , f coll (X = 0, ξ , η, t) is the distribution function of molecules colliding against the interface, ξx is the molecular velocity component normal to the interface, ξ y and ξz are the tangential components. In the first equation of Eq. (3.14) and all equations appearing below, the integration with respect to η is taken over the range 0 η < ∞. Equation (3.13) is the spatially one-dimensional version of Eq. (2.103).3

*∞ Γ (n/2 + 1) is the gamma function whose definition√is given by Γ (x) = 0 e−t t x−1 dt for x > 0 [27]. This has characteristics as follows: Γ (1/2) = π , Γ (1) = 1, and Γ (x + 1) = xΓ (x).

2 3

The molecular velocity has three components even in spatially one-dimensional problems.

76

3 Methods for the Measurement of Evaporation and Condensation Coefficients

3.1.3 KBC Expressed by Net Mass Flux Measured at the Interface The mass fluxes, Jout , Jevap , and Jcoll at the interface are given by Jout = (ξx − vw ) f out dξ dη, ξx >vw

(3.15)

Jevap =

ξx >vw

(ξx − vw ) f evap dξ dη,

(3.16)

Jcoll = −

ξx
(ξx − vw ) f coll dξ dη,

(3.17)

where f evap (X = 0, ξ , η, t) is the distribution function for spontaneously evaporating liquid molecules across the interface. Using Eqs. (3.15), (3.16), and (3.17), we obtain Jref = Jout − Jevap ,

(3.18)

Jcnds = Jcoll − Jref .

(3.19)

Here, defining the net mass flux of evaporation, M, across the interface as M = Jevap − Jcnds ,

(3.20)

and we can rewrite it with the use of the definitions of αe and αc given by Eq. (3.1) as RTw ∗ M = (αe ρ − αc σ ) , (3.21) 2π where σ is defined by the first equation of Eq. (3.14). The mass flux M is defined as a positive quantity for evaporation and it is approximately equal to the liquid mass, −ρ L vw , subtracted from the interface per unit time and unit area by the evaporation, where ρ L is the liquid density and the fluid velocity inside the liquid is neglected because it is quite small compared with the interface velocity in the situation considered in Sect. 3.2 (see Fig. 3.4). Equation (3.21) is identical with Eq. (2.143). Finally, we obtain RTw ∗ . (3.22) ρ L vw = (αc σ − αe ρ ) 2π Making use of Eq. (3.22) allows us to eliminate αe and αc from Eq. (3.13) and leads to 2π out f (X = 0, ξ , η, t) = σ − ρ L vw fˆtr fˆint , (ξx > vw ). (3.23) RTw

3.1

Review of αe , αc , KBC, and Gaussian–BGK Boltzmann Equation

77

Once the net mass flux of condensation ρ L vw , i.e., the left-hand side of Eq. (3.22), is given as a function of time in any experimental ways, the Gaussian–BGK Boltzmann equation can be solved with the KBC (3.23) together with Eq. (3.14) without knowledge of the two coefficients in advance.

3.1.4 Gaussian–BGK Boltzmann Equation in Moving Coordinate System We will rewrite the Gaussian–BGK Boltzmann equation (2.93) in the moving coordinate system whose origin moves with the speed vw , as shown in Fig. 3.3. Let X be the coordinate stemming from the moving interface into the vapor. If X 0 is the coordinate measured from initial position of the interface, the relation between X 0 and X is given by t X0 = X + vw (t)dt. (3.24) 0

With the use of Eq. (3.24), the differentiation of the distribution function f (X, ξ , η, t) with respect to the time t in the fixed coordinate system can be represented as ∂f ∂f − vw , ∂t ∂X

(3.25)

and the differentiation of f with respect to X 0 is rewritten as ∂ f /∂ X . Therefore, the spatially one-dimensional version of Eq. (2.93) can be given in the moving coordinate system as ∂f p ∂f + (ξx − vw ) = [G( f ) − f ], ∂t ∂X μ(1 − ν + θ ν)

(3.26)

where p is the vapor pressure, μ is the viscosity coefficient of the vapor, ν and θ are constant values, and the function G( f ) is given by Eq. (2.94). The macroscopic variables such ρ, vi (i = x, y, z), and T etc. are given in Eqs. (2.96), (2.97), (2.98), and (2.99).

Vapor

Interface

Liquid

Fig. 3.3 The relation between the fixed coordinate and the coordinate moving with the velocity vw

78

3 Methods for the Measurement of Evaporation and Condensation Coefficients

3.2 Shock Tube Method for Measurement of Condensation Coefficient 3.2.1 Principle of Shock Tube Method As explained in Sect. 3.1, the Gaussian–BGK Boltzmann equation (3.26) can be solved together with Eq. (3.23). The value of αc can be deduced from Eq. (3.22) with the use of σ, ρ ∗ , Tw , and αe , once the net mass flux of condensation ρ L vw is given as a function of the time in any experimental ways. The value of αe in Eq. (3.22) is evaluated by the MD simulation, and its use will be justified by experiments made in Sect. 3.6. The nonequilibrium state can be realized in the region between the vapor–liquid interface of a thin liquid film formed on the shock tube endwall and a shock wave reflected there [5–9, 18, 19, 23, 24, 26]. Figure 3.4 shows the propagation process of a shock wave in a vapor advancing toward and reflecting from the shock tube endwall. The part shown by hatching is the endwall, on which a thin adsorbed liquid film exists initially, the right side of the adsorbed liquid film on the endwall, i.e., in the test section of the shock tube is filled with a test vapor. The existence of the adsorbed liquid film will be demonstrated in Sect. 3.5. Just at the instant when the incident shock wave is reflected at the endwall surface covered with the adsorbed liquid film, the pressure and density of the vapor at the

Liquid Film Thermal Boundary Layer

Shock Tube Endwall

Time

Interface Vapor Flow

Trajectory of Reflected Shock Wave Trajectory of Incident Shock Wave

Vapor Initially Adsorbed Liquid Film

Fig. 3.4 The propagation process of the shock wave in the vapor advancing toward and reflecting from the shock tube endwall and the subsequent growth of the liquid film on the endwall. The presence of the initially adsorbed liquid film on the endwall is crucial for the uniform liquid film growth

3.2

Shock Tube Method for Measurement of Condensation Coefficient

79

interface increase rapidly in the time scale of mean free time of vapor molecules and these become almost constant values, respectively. The vapor temperature at the interface, on the other hand, changes little because the thermal process of the very thin liquid film is governed by the thick endwall with a large heat capacity compared with that of the vapor. The initial vapor pressure is set at an appropriate value so that the interfacial vapor pressure after the shock wave reflection may become higher than the saturated vapor pressure at the liquid film temperature. The Knudsen layer is formed in the vapor near the interface, and the thermal boundary layer also develops outside the Knudsen layer with the lapse of time. As a result, the vapor becomes supersaturated at the interface, and it begins to further condense on the initial liquid film. The liquid film grows with the lapse of time. Figure 3.5 shows changes in thermodynamic states of the vapor exposed to both the incident and successively reflected shock waves. The figure is depicted on the pressure-specific volume plane. The initial state of the vapor prior to the arrival of the incident shock wave is denoted by the number 1. The state of the main stream changes from State 1 to State 2 through the two stages of almost adiabatic compression4 ; first by the incident shock wave and subsequently by the reflected shock wave. However, the state at the interface changes, in the time scale of mean free time of vapor molecules, from State 1 to State 2W; the vapor pressure at State 2W is nearly equal to the pressure at State 2. State 2W is a supersaturated one under which

C Critical Point

Pressure

Liquid Line Saturation Line

2W State of Vapor at Interface

2 State of Main Stream

1 Initial State

Specific Volume

Fig. 3.5 Changes of thermodynamic states of the vapor. The dashed curves are isothermal lines. The change of the vapor from State 1 to State 2W proceeds along the isothermal line, on which the temperature is equal to the initial one. The saturated pressure at the temperature on the isothermal line is lower than the pressure of State 2, and this induces the condensation. The change from State 1 to State 2 is almost adiabatic, which is shown by the dash-dotted line in the figure

4 The shock wave is essentially the non-adiabatic process, which is accompanied by the increase of entropy. However, the increase of entropy of weak shock wave is proportional to the cubic power of shock strength, and hence the adiabatic approximation is often valid for weak shock waves.

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3 Methods for the Measurement of Evaporation and Condensation Coefficients

the condensation takes place, resulting in the liquid film which is formed uniformly on the initially adsorbed liquid film. The vapor and liquid film system mentioned above provides us the simplest and most ideal circumstance for the measurement of the condensation coefficient αc at the vapor–liquid interface.

3.2.2 Characteristics of Film Condensation at Endwall behind Reflected Shock Wave Before proceeding to the detailed analysis of the film condensation on the shock tube endwall, we shall understand typical characteristics of the liquid film growth on it. According to works of Fujikawa et al. [5, 7], which theoretically analyzed the liquid film growth on the endwall behind the reflected shock wave, the growth rate of the liquid film can be described by a Volterra integral equation of the second kind as5 t dδ(t) dδ(t˜)/dt˜ β2 = β1 − √ dt˜. (3.27) dt π 0 t − t˜ Here, δ is the thickness of liquid film, t is the time, and p∞ ( p∞ − p0∗ ) β1 = φ(α)ρ L p0∗

2 , RT0

p∞ ( p∞ − p0∗ + 2Ap∞ )L β2 = φ(α)k S p0∗ √ 1−α , φ(α) = 2.13204 + 2 π α

DS

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

, ⎪ 2RT03 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(3.28)

where T0 is the initial temperature of liquid film and vapor, p0∗ is the initial vapor pressure which is assumed equal to the saturated vapor pressure at T0 ,6 p∞ is the vapor pressure far from the vapor–liquid interface behind the reflected shock wave, ρ L is the density of liquid film, D S is the thermal diffusivity of endwall, k S is the thermal conductivity of endwall, L is the latent heat of condensation, R is the gas constant per unit mass, and A = bT0 /(c+T0 )2 where b and c are the constant values in Antoine’s equation for the saturated vapor pressure, Eq. (1.7).

5 6

This is the same equation as Eq. (1.11) except for definitions of β1 and β2 .

This assumption is just made for simplicity of the problem in question. We will deal with a general problem for the case where the initial pressure is not equal to p0∗ , in the next subsection.

3.2

Shock Tube Method for Measurement of Condensation Coefficient

81

The solution of Eq. (3.27) can be obtained to be [4] √ dδ(t) = β1 exp(β22 t)erfc(β2 t). dt

(3.29)

Integrating Eq. (3.29) with respect to the time, we obtain δ(t) =

√ √ 2 β1 2 β t)erfc(β t) + t − 1 + δ0 , exp(β √ 2 2 2 π β22

(3.30)

where δ0 is the initial thickness of liquid film. The temperature at liquid film surface, Tw (t) = TL (X = 0, t), and that at the endwall surface, TS (X = −δ, t), are given by Tw (t) = TL (X = 0, t)

√ √ β1 ρ L L D S ∼ ∼ 1 − exp(β22 t)erfc(β2 t) . = TS (X = −δ, t) = T0 + β2 k S

(3.31)

Further analysis of Eqs. (3.30) and (3.31) leads to the following approximate expressions for two cases: √ (1) for β2 t 1; δ(t) = β1 t, Tw (t) = T0 +

2β1 ρ L L kS

(3.32) DS t, π

(3.33)

√ (2) for β2 t 1; β1 2 β1 √ t − 2 + δ0 , δ(t) = √ π β2 β2 √ ρ L L D S β1 Tw = T0 + (= const.), kS β2

(3.34)

(3.35)

where we should notice, for the following discussion, that β1 and β1 /β22 are dependent on φ(α), while β1 /β2 is independent of φ(α), as discussed in Sect. 1.2. We can understand that the liquid film growth is drastically influenced by φ(α)√and the elapsed time t from the instant of the stepwise change of the state. For β2 t 1, the liquid film thickness changes in proportion to the time t and it also depends on √ φ(α), while for β2 t 1 it changes in proportion to the square root of the time, and the growth behavior of liquid film becomes independent of φ(α) as the time lapse and becomes strongly dependent on thermophysical properties of the vapor and liquid. Equations (3.32), (3.33), (3.34), and (3.35) are similar to but different

82

3 Methods for the Measurement of Evaporation and Condensation Coefficients

from Eqs. (1.17), (1.18), (1.19), and (1.20) in the point that the heat conduction process in the present case is controlled by thermophysical properties of the shock √ tube endwall, such as D S and k S . Defining the transition time τt to be β2 τt = 1, we obtain7 τt =

2RT03 DS

φ(α)k S p0∗ p∞ ( p∞ − p0∗ + 2Ap∞ )L

2 .

(3.36)

Comparing this equation with Eq. (1.21), we notice the difference between D L and k L in Eq. (1.21) and D S and k S in Eq. (3.36). In the latter case, the temperature rise of liquid film surface is controlled by the heat conduction of the shock tube endwall because the liquid film is so thin compared with the endwall. Evaluating τt for the small pressure change by 5% from the saturated vapor pressures at 290 K for methanol and water, we obtain, e.g., for α = 1 (the shortest case): (i) τt = 0.6 µs for methanol, (ii) τt = 6 µs for water. The liquid film growth is sensitive to the change of φ(α), i.e., α, as shown in Eq. (3.32) before τt , while after τt its dependence on φ(α) becomes smaller and smaller as the time elapses, as shown in Eq. (3.34). This result implies that the growth of the thickness of liquid film should be measured in a sufficiently short time (≈ τt ) after the shock wave is reflected at the endwall. The temperatures of liquid film and endwall surfaces change little, and therefore the effect of the temperature rise of liquid film surface on the measurement of α can be ignored,8 when we measure the liquid film thickness in the time scale of τt .

3.2.3 Mathematical Modeling of Film Condensation on Shock Tube Endwall As shown in Fig. 3.6, a steady shock wave with plane front is formed at a position far from the shock tube endwall covered with the initially adsorbed liquid film in the vapor and it propagates toward the endwall. In front of the incident shock wave with Mach number Ms, the distribution function is given by ρ0 fˆtr fˆint for the stationary vapor with the temperature T0 and the density ρ0 . Behind the shock wave, the distribution function is also given by ρb fˆtr fˆint for the flowing vapor with the temperature Tb , the density ρb , and the velocity vb . These quantities both in

The transition time τt is the characteristic time when the growth of liquid film thickness changes √ from the t-proportion manner to the t-proportion one, as discussed in detail in Sect. 1.2.

7

8 In Sect. 3.6, we carry out rigorous and complete computations of the liquid film growth and vapor flow without ignoring the temperature rise of liquid film surface.

3.2

Shock Tube Method for Measurement of Condensation Coefficient

83

Shock Tube Endwall (Optical Glass) Initially Adsorbed Liquid Film Interface

Incident Shock Wave (Mach Number Ms)

Polyatomic Vapor O

T0

T0

Tb , p b , ρ b, vb

X

T0 , p 0 , ρ0

Fig. 3.6 The schematic diagram of the incident shock wave advancing toward the shock tube endwall covered with the initially adsorbed liquid film. Quantities with the subscript 0 denote their initial values and those with b denote their values behind the incident shock wave

front of and behind the shock wave can be connected by the Rankine–Hugoniot relations [10]: ρb (γ + 1)Ms2 = , ρ0 2 + (γ − 1)Ms2

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

Tb (2γ Ms2 − γ + 1)[2 + (γ − 1)Ms2 ] = ,⎪ ⎪ T0 (γ + 1)2 Ms2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ vb ρ0 ⎪ ⎪ √ =1− . ⎭ ρb γ RT0

(3.37)

The Gaussian–BGK Boltzmann equation and the KBC are respectively given by Eqs. (3.26) and (3.23) together with Eq. (3.14). We also solve the heat conduction equations for the liquid film and endwall for precise estimation of the temperature of liquid film surface. We assume that the phenomenon considered is one-dimensional. However, we should discuss the validity of the one-dimensional assumption, because the flow field behind the reflected shock wave may become complicated due to the interaction between the reflected shock wave and the boundary layer developed by the incident shock wave on the shock tube sidewall. Fujikawa et al. [8] have made numerical simulations of the reflection process of a shock wave on the shock tube endwall in water vapor and of the growth process of liquid films formed on both the endwall and sidewall. The set of two-dimensional Navier–Stokes equations for the vapor and the heat conduction equations for both the liquid film and shock tube walls have been solved under the boundary conditions (1.1) and (1.2) by finite difference methods. It has been shown that the flow field in the shock wave reflection region becomes very complicated due to the interaction of the reflected shock wave with

84

3 Methods for the Measurement of Evaporation and Condensation Coefficients

the boundary layer on the sidewall. But, there is no large difference near the center of the shock tube between the two-dimensional result of liquid film growth and the one-dimensional one. This suggests that the measurement of the liquid film thickness should be carried out near the shock tube axis.

3.2.4 Boundary Condition at Infinity in Vapor The boundary condition at infinity in the vapor is given by [18] f = ρb fˆtr (Tb ) fˆint (Tb ),

(3.38)

where (ξx − vb )2 + ξ y2 + ξz2 1 , exp − fˆtr (Tb ) = 2RTb (2π RTb )3/2 fˆint (Tb ) =

2/n 1 η . exp − n/2 RTb Γ (n/2 + 1) (RTb )

(3.39)

(3.40)

3.2.5 Heat Conduction in Liquid Film and Shock Tube Endwall We need to solve the Gaussian–BGK Boltzmann equation for the vapor together with the heat conduction equations for the liquid film and shock tube endwall [2]: ∂ 2 TL ∂ TL , = DL ∂t ∂ X 02

(3.41)

∂ TS ∂ 2 TS = DS , ∂t ∂ X 02

(3.42)

where t is the time, TL and TS are respectively the temperatures of the liquid film and shock tube endwall, D L and D S are the thermal conductivities of the liquid film and shock tube endwall. The space coordinate X 0 shown in Fig. 3.3 is adopted in Eqs. (3.41) and (3.42). The boundary conditions for Eqs. (3.41) and (3.42) are given by [18] ∂ TL −k L = (L − h) (ξx − vw ) f dξ dη ∂ X0 1 2 2/n ξ +η f dξ dη, + (ξx − vw ) 2 i

(X 0 = δ),

(3.43)

3.2

Shock Tube Method for Measurement of Condensation Coefficient

T L = TS ,

kL

(X 0 = 0),

∂ TL ∂ TS = kS , ∂ X0 ∂ X0 TS = T0 ,

(X 0 = 0),

(X 0 = −∞),

85

(3.44)

(3.45)

(3.46)

where ξi2 = ξx2 + ξ y2 + ξz2 according to the Einstein summation convention in Appendix A.2 at the end of this book, L is the latent heat of condensation per unit mass, h is the enthalpy of the vapor per unit mass, k is the thermal conductivity, and the subscripts L and S denote the liquid film and shock tube endwall, respectively. Equation (3.43) holds at the interface between the vapor and liquid film,9 Eqs. (3.44) and (3.45) hold at the boundary between the liquid film and shock tube endwall, and Eq. (3.46) holds at infinity in the shock tube endwall.

3.2.6 Initial Conditions We impose the following initial condition for the stationary vapor in front of the incident shock wave [18]: f = ρ0 fˆtr (T0 ) fˆint (T0 ),

(3.47)

ξx2 + ξ y2 + ξz2 1 ˆ , f tr (T0 ) = exp − 2RT0 (2π RT0 )3

(3.48)

where

fˆint (T0 ) =

2/n 1 η . exp − √ RT0 Γ (n/2 + 1) (RT0 )n

(3.49)

The initial conditions for the liquid film and shock tube endwall are also given by TL = T0 ,

(0 ≤ X 0 ≤ δ),

(3.50)

TS = T0 ,

(−∞ ≤ X 0 ≤ 0).

(3.51)

9 In Eq. (3.43), some small terms are neglected, which are the transport of liquid kinetic energy across the interface and works by pressure and viscous stress in the liquid. The precise boundary condition will be given in Chap. 5.

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3 Methods for the Measurement of Evaporation and Condensation Coefficients

3.3 Shock Tube 3.3.1 Schematic and Performance of Shock Tube Figure 3.7 shows a horizontal type of shock tube used for the experiment, in which the optical interferometer is incorporated for the measurement of liquid film thickness [9, 18, 19]. The length of the low pressure section, i.e., test section, is 2830 mm, and the length of the high pressure section is 2600 mm. Two sections are made of circular pipes with inner diameter 74.3 mm. They are connected coaxially, and a thin aluminum diaphragm is sandwiched between them. The endwall of the test section consists of a circular quartz plate of radius 20 mm and thickness 15 mm. The surface of the plate is optically polished at the flatness level of λ/20 (λ = 632.8 nm) to prevent the diffused reflection of a laser beam for the optical measurement and to make the laser-shed region of the liquid film as uniform as possible. First of all, the test section is evacuated by the vacuum pumps shown in Fig. 3.7, and then filled with the test vapor so that the shock wave can propagate in pure vapor. The attained vacuum level of the test section is 1.5 × 10−3 Pa, and the leakage speed of this section is about 0.01 Pa/min. Pure methanol or water vapor is then introduced into the test section from the vapor tank. It takes about 5 min in order to introduce the vapor into the test section, rearrange, and calibrate the optical system, so the final pressure of noncondensable gases (mainly air) just before the run of experiment may be about 0.05 Pa. For test gases such as methanol and water vapors, this percentage of noncondensable gases contained is less than 1.7 × 10−2 % at room temperature. This level of noncondensable gases in the vapor has no influence with the liquid film growth, as will be shown in the next subsection.

High Pressure Section

Low Pressure Section

Optical Interferometer Beam Splitter

Baratron Pressure Gauge

Reference Beam

Ionization Gauge Thermocouple

Optical Glass

Lens

Photodiode

Diaphragm Kistler Pressure Gauge

Physical Beam

Charge Amplifier Diffusion Pump Time Interval Analyzer Rotary Pump Driver Gas Tank [N2]

Vapor Tank Oscilloscope Personal Computer

Fig. 3.7 The shock tube and optical interferometer

He-Ne Laser

3.3

Shock Tube

87

Before the start of each experiment, the initial pressure of the vapor is measured by Baratron pressure gauge (type 122A, MKS) and initial temperatures of the vapor and the endwall are measured by thermocouples with the accuracy of 0.1◦ C. The nitrogen gas, driver gas, is introduced into the high pressure section, and the diaphragm is naturally ruptured by the pressure difference between the driver gas and the vapor. Then, a shock wave is generated and propagates toward the endwall in the vapor of low pressure section. The Mach number of this incident shock wave is evaluated from the distance (= 1000 mm) between two pressure gauges (type 701A, Kistler) and the passage time of the shock wave between them. All data except the temperature are recorded in digital storage oscilloscopes and then processed by computers.

3.3.2 Effect of Noncondensable Gases on Liquid Film Growth The existence of a small amount of noncondensable gases such as air in a test vapor may have a great influence on accuracy in the measurement of the evaporation and condensation coefficients, because the noncondensable gases accumulate at the interface due to the vapor flow induced by the condensation and prevent the vapor from condensing. It is important to make clear by numerical simulations the effect of noncondensable gases on the liquid film growth on the shock tube endwall. Figure 3.8 shows the time change of the thickness of a liquid film growing on the shock tube endwall in a mixture of methanol vapor and air; the initial mass fraction cg∞ of air in the mixture is changed as 0, 0.1, 1, and 5% [23, 25]. The abscissa is the square root of the time. The other initial conditions are: temperature = 293.15 K, 100 Liquid Film Thickness (nm)

cg∞ = 0 % 0.1 %

75

1%

50

5%

25

0

0

2.5

5.0 7.5 [t (μs)]1/2

10.0

Fig. 3.8 The time change of the liquid film thickness on the shock tube endwall: cg∞ = 0, 0.1, 1, and 5%

88

3 Methods for the Measurement of Evaporation and Condensation Coefficients

vapor pressure = 1.1095×104 Pa (saturated vapor pressure at 293.15 K), liquid film thickness = 10 nm, incident shock Mach number = 1.1, and α = 0.15. This value of α is the averaged experimental one at the time when this analysis was made [24]. The growth rate of liquid film is greatly influenced by an initial amount of air in the mixture. For example, the growth of liquid film is not significantly affected by air for cg∞ = 0.1%, while it is greatly suppressed for 5%. Also, the growth is not influenced at all by any air amounts at the time stage during about 1.5 µs after condensation started. These results suggest that the liquid film growth in the mixture when the initial gas concentration less than 0.1% can be regarded as that in the pure vapor with negligible errors and that the measurement of liquid film thickness should be made as early as possible after the start of condensation. As a whole, the growth behavior of liquid film in the mixture is very similar to that of the pure vapor. That is, there exists the temporal transition phenomenon on the growth of the liquid film; the film grows approximately in proportion to the time in early time stages, and after a transition period, it does asymptotically in proportion to the square root of the time. The transition time is delayed by the inclusion of a small amount of air.

3.3.3 Effect of Association of Molecules on Vapor State A vapor is not always in a monomer state and its state depends on its species and temperature. Vapor molecules can aggregate to form dimers, trimers, etc. as the temperature becomes lower [9, 20–22]. This phenomenon is a kind of association of molecules, and the association in an initial state of the vapor may have an influence on the successive condensation process after the shock wave reflection. We shall here discuss the association of molecules in the initial vapor state. Methanol and water vapors will be treated as chemically reactive mixtures consisting of their monomers and dimers as 2CH3 OH (CH3 OH)2 ,

(3.52)

2H2 O (H2 O)2 .

(3.53)

The degree of association, β, of the vapor is given by [22] K (T ) 4(1 − β)2 = , β(2 − β) p

(3.54)

especially for small values of β ( 1) 2p β∼ , = K (T )

(3.55)

3.4

Optical Interferometer

89

where p is the total pressure of a mixture and K (T ) is the equilibrium constant of the mixture for a fixed temperature. The constant K (T ) is, for example [21, 22], K = 2.54 × 106 Pa for methanol vapor at 290 K, K = 2.21 × 106 Pa for water vapor at 290 K.

& (3.56)

For typical conditions in initial pressures of methanol and water vapors (see Fig. 3.14), the values of β can be estimated to be, respectively, β = 0.3% for methanol vapor at 4, 000 Pa, β = 0.03% for water vapor at 300 Pa.

& (3.57)

We can understand that the degree of association is very small for both vapors in the typical conditions.10 Furthermore, we have confirmed that the initial degree of the association in the shock tube is kept at the same value even after the shock wave reflection at the shock tube endwall. That is, the association degree is frozen. It can therefore be concluded that the molecular association does not affect the condensation process concerned.

3.4 Optical Interferometer 3.4.1 Theory of Optical Interferometer The time change of liquid film thickness can be obtained by the measurement of the energy reflectance of a light beam from the optically transparent liquid film system, because the growing liquid film, together with the shock tube endwall made of the optical glass (quartz plate), forms a kind of interferometer [6, 9, 18, 19, 23, 24, 26]. Figure 3.9 shows a multi-layer model of transmission, refraction, and reflection of the light shed from the right side with an incident angle in the atmosphere in the configuration of the optical system shown in Fig. 3.7. The thickness and refractive index of the optical glass are respectively d2 and n 2 . The layer indicated by the subscript 1 is the liquid film which grows on the glass in the shock wave reflection region; the thickness = d1 and the refractive index = n 1 . The thickness = d1 is a function of the time, because it changes due to the condensation as the time lapse. The refractive index of the vapor is n 0 . The layer indicated by the subscript 3 is the liquid film which is formed by adsorption of water vapor molecules in the atmosphere; the thickness = d3 and the refractive index = n 3 . The existence of this liquid film can easily be demonstrated by heating up the glass surface facing the

However, for acetic acid vapor (CH3 COOH), the value of β is strongly dependent on the temperature [20]; it is about 80% of the vapor in the association state at 300 K, and about 20% in the association state at 350 K.

10

90

3 Methods for the Measurement of Evaporation and Condensation Coefficients n0 Vapor d1

i1

n1 Liquid Film

i2

d2

d3

n2 Optical Glass

i3

n3 Liquid Film n4

Air

Light Beam

Fig. 3.9 The multi-layer model of transmission, refraction, and reflection of the light in the optical system in the shock tube shown in Fig. 3.7

atmosphere with a hair dryer. The atmosphere is indicated by the subscript 4; the refractive index = n 4 . For normal incidence of the light beam shed to the optical glass (i 1 = i 2 = i 3 = 0), the energy reflectance, I (t), is given by [26] I (t) = 1 −

(r02 − 1)(r12 − 1)(r22 − 1)(r32 − 1) , D(t)

(3.58)

where r0 =

n0 − n1 , n0 + n1

r1 =

n1 − n2 , n1 + n2

r2 =

n2 − n3 , n2 + n3

r3 =

n3 − n4 , n3 + n4

(3.59)

D(t) = 1 + (r0 r1 )2 + (r1 r2 )2 + (r2r3 )2 + (r0 r2 )2 + (r0 r3 )2 + (r1 r3 )2 + (r0 r1 r2r3 )2 + 2r2r3 (1 + r02 )(1 + r12 ) cos δ3 + 2r1r2 (1 + r02 )(1 + r32 ) cos δ2 + 2r0r1 (1 + r22 )(1 + r32 ) cos[δ1 (t)] + 2r0r2 (1 + r32 ){cos[δ1 (t) + δ2 ] + r12 cos[δ1 (t) − δ2 ]} + 4r0r1 r2 r3 {cos[δ1 (t) + δ3 ] + cos[δ1 (t) − δ3 ]} + 2r1r3 (1 + r02 )[cos(δ2 + δ3 ) + r22 cos(δ2 − δ3 )] + 2r0r3 {cos[δ1 (t) + δ2 + δ3 ] + r22 cos[δ1 (t) + δ2 − δ3 ] + r12 r22 cos[δ1 (t) − δ2 + δ3 ] + r12 cos[−δ1 (t) + δ2 + δ3 ]},

(3.60)

3.4

Optical Interferometer

91

4π n 1 d1 (t) , λ

δ1 (t) =

δ2 =

4π n 2 d2 , λ

δ3 =

4π n 3 d3 , λ

(3.61)

where t is the time. It is reasonable to assume that the refractive indices, i.e., n 0 , n 1 , n 2 , n 3 , and n 4 , are constant values, as will be demonstrated in Sect. 3.5.3. Numerical simulations made systematically have shown that the energy reflectance I (t) is strongly dependent on the optical glass thickness d2 and the liquid film thickness d3 in the atmosphere as well as the liquid film thickness d1 to be measured. The liquid film thickness in the atmosphere, d4 , is not controllable at all unless the humidity of the optical system is kept constant, and the optical glass thickness can not be measured so accurately in the wavelength (nanometer) scale. This result means that the use of Eq. (3.58) is not suitable for the measurement of the liquid film thickness d1 . Furthermore, Eq. (3.58) is not easily tractable for the deduction of d1 (t). Instead of using Eq. (3.58), we shall seek the simpler way for deduction of d1 (t). Figure 3.10 shows a simple model in which the interference of light is taken into account only in the liquid film with the subscript 1. For the normal incidence of the light beam shed to the optical glass (i 1 = 0), the energy reflectance I s (t) is given by [26] I s (t) = 1 −

4n 2 n 21 n 0 n 21 (n 2 + n 0 )2 − (n 22 − n 21 )(n 21 − n 20 ) sin2 [δ1 (t)/2]

,

(3.62)

where the superscript s denotes the simple model. Equation (3.62) has the maximum and minimum as follows: ⎫ n2 − n0 2 ⎪ ⎪ = at δ1 = 2kπ, (k = 0, 1, 2, . . .), ⎪ ⎬ n2 + n0 2 2 ⎪ n1 − n2n0 ⎪ = at δ1 = (2k + 1)π, (k = 0, 1, 2, . . .). ⎪ ⎭ 2 n1 + n2n0

s Imax s Imin

(3.63)

Figure 3.11 shows the comparison of Eqs. (3.62) and (3.58) for one cycle of oscillation of energy reflectance; the solid line for Eq. (3.58) and the dashed line for Eq. (3.62) [26]. The abscissa is the film thickness d1 (t), and the ordinate is the normalized energy reflectance [I (t) − Imin ]/(Imax − Imin ) for Eq. (3.58) and n0 Vapor d1

Light Beam

i1

n1 Liquid Film n2 Optical Glass

Fig. 3.10 The optical configuration for the simple model

3 Methods for the Measurement of Evaporation and Condensation Coefficients Normalized Energy Reflectance

92

Eq. (3.58)

Eq. (3.62)

Variation of Liquid Thickness (nm)

Fig. 3.11 The comparison between Eqs. (3.58) and (3.62)

s ]/(I s 2 [I s (t) − Imin max − Imin ) for Eq. (3.62), respectively. In the evaluation of Eq. (3.58), we suppose that the two liquid films on both glass surfaces are methanol (n 1 = n 3 = 1.3282 for λ = 632.8 nm at 20◦ C), the vapor is methanol vapor (n 0 = 1.00006 for the saturated vapor at 20◦ C), and the glass is quartz (n 2 = 1.51481), and the refractive index n 4 of the atmosphere is 1.00027 at 20◦ C. Two curves with the largest difference between Eqs. (3.58) and (3.62) are illustrated in the figure. It is found that Eq. (3.62) for the simple model is in quite good agreement with Eq. (3.58) for the realistic model and that the maximum relative difference between Eqs. (3.58) and (3.62) is less than 4%. Therefore, the simplification is possible with acceptable error bounds. Equation (3.62) will henceforth be used to deduce the liquid film thickness from experimental energy reflectance data.

3.4.2 Method of Optical Data Analysis The energy reflectance I s (t) is measured by a photodiode which is used in its linear characteristic region, and the output of the photodiode will be expressed by K (t) with the unit of voltage [23, 24, 26]. The measured energy reflectance K (t) will be normalized as R(t) =

K (t) − K min , K max − K min

(3.64)

where K max and K min are the maximum and minimum of the measured energy s s ]/(Imax − reflectance, respectively. Equation (3.64) can be set equal to [I s (t) − Imin s Imin ), when the photodiode is used in the linear characteristic region as it is actually done. Taking the origin of time just at the instant of the reflection of the shock wave, i.e., the start of condensation, the time change of the liquid film thickness is given by

3.5

Properties of Adsorbed Liquid Film on Optical Glass Surface

Δd1 (t) =

λ 4π n 1

cos−1

s a+ Imin − a− + a+ ΔI s · R(t) − d10 , s − ΔI s · R(t) 1 − Imin

93

(3.65)

where ⎫ n 21 (n 2 ± n 0 )2 + (n 21 ± n 2 n 0 )2 ⎪ ⎪ ,⎬ a± = (n 22 − n 21 )(n 21 − n 20 ) ⎪ ⎪ ⎭ s s − Imin . ΔI s = Imax

(3.66)

The initial value R(t = 0) of the normalized energy reflectance can be known from the experimental R(t) datum, and d1 (t = 0) is not the real thickness of the liquid film at t = 0, because it appears from the fit between the experimental normalized s s s ]/(Imax − Imin ). The variation reflectance R(t) and the theoretical one [I s (t) − Imin Δd1 (t) in time of the thickness after the condensation onset can be determined by Eq. (3.65). In the experiment, the light source is He-Ne laser beam (50 mW) with the wavelength of λ = 632.8 nm; n 0 = 1.00006 (methanol and water vapors), n 1 = 1.3282 for methanol, n 1 = 1.3311 for water, and n 2 = 1.51481. In the measurement of the light reflectance, the following special device is made to eliminate systematic noises from the light reflectance. As shown in Fig. 3.7, the light beam from the light source is divided into two beams; one is the physical beam containing information of the liquid film thickness, and the other is the reference one which is to be subtracted from the light reflectance of the physical beam in order to eliminate the noises. The physical and reference beams are detected by two photodiodes, respectively. The response time of electronic circuits for the photodiodes is less than 0.4 µs and the time interval of data acquisition is 2 ns. These performances are sufficient for accurate measurements of the liquid film thickness, because a typical rise time needed to measure the growth rate of the liquid film is about 10 µs for both methanol and water vapors (see Fig. 3.14).

3.5 Properties of Adsorbed Liquid Film on Optical Glass Surface 3.5.1 Treatment of Optical Glass The optical glass surface must be specially treated in order to produce a uniform liquid film on it. The glass is first dipped for about 30 min in a solution consisting of sulfuric acid and bichromate or a special solution of glass washing, then it is washed by distilled water and is dipped in a boiling test liquid for about 5 min. The glass is finally wiped many times by washed gauze containing the test liquid. It is known that a permanently adsorbed liquid film of about 10 nm in thickness exists on the glass surface even in vacuum unless the glass is treated by special methods, e.g.,

94

3 Methods for the Measurement of Evaporation and Condensation Coefficients

heating up to about 800 K for water [29]. The permanently adsorbed liquid film of the test liquid therefore exists on the glass surface, when the glass is treated by the way mentioned above, before it is mounted into the endwall of the shock tube.

3.5.2 Thickness of Temporarily Adsorbed Liquid Film We discussed the permanently adsorbed liquid film on the optical glass surface in the above subsection. However, we cannot recognize it in usual ways and also need not to do it. There exists another adsorbed liquid film which is temporarily produced on the glass surface mounted into the endwall, when a test vapor is introduced into the test section of the shock tube. The presence of this temporarily adsorbed liquid film on the glass surface is crucial for the successive uniform growth of the liquid film after an incident shock wave was reflected at the endwall. Figure 3.12a, b show the pressure dependence of the liquid films adsorbed on the glass surface of the shock tube endwall for methanol and water vapors, respectively [23, 24]. The abscissas are indicated by both the absolute vapor pressure and the relative humidity. We find that molecules of the vapors deposit on the glass surface even under the conditions where their pressures are lower than the saturated ones. The higher are the pressures, the thicker become the adsorbed liquid films.

Thickness of Adsorbed Liquid Film (nm)

(a)

(b)

6 Saturated Vapor Pressure = 2.145 kPa Temperature = 291.7 K

Saturated Vapor Pressure = 10.379 kPa Temperature = 291.9 K

5

2

4 3 1 2 1 0

0

0

1

2 3 4 5 6 7 8 9 10 Vapor Pressure (kPa) 20 40 60 80 Relative Humidity (%)

100

0

0

0.5 1.0 1.5 Vapor Pressure (kPa)

0

20 40 60 80 Relative Humidity (%)

2.0

100

Fig. 3.12 The thickness of temporarily adsorbed liquid film and its pressure dependence: (a) methanol vapor and (b) water vapor

3.5

Properties of Adsorbed Liquid Film on Optical Glass Surface

95

3.5.3 Refractive Index of Initially Adsorbed Liquid Film

10 26˚C

1.340

Refractive Index

1.320

8

6 1.300

4 Thickness of Adsorbed Liquid Film

2

Refractive Index

Thickness of Adsorbed Liquid Film (nm)

Figure 3.13 shows the relation between the thickness and the refractive index of the initially adsorbed liquid film of water at 26◦ C against p/ p ∗ , where p is the vapor pressure and p∗ is the saturated vapor pressure [17]. The refractive index rises sharply in the range between p/ p ∗ ∼ = 0.45 and 0.6, while the thickness remains almost constant (∼ = 0.7 nm). This suggests a gradual change in the structure of the adsorbed liquid film, starting from some island structure to a continuous molecular film. Above p/ p ∗ ∼ = 0.6, the thickness rises rapidly while the refractive index remains equal to 1.33, suggesting that the liquid film is the bulk liquid in nature. This figure also demonstrates that the liquid film with about 1.3 nm in thickness has the bulk liquid nature in the refractive index. It should be noted that Fig. 3.13 is the relation between the liquid film with a constant thickness and its refractive index. As will be shown in Sect. 3.6.3, the water vapor pressure increases up to eleven times the initial pressure at about 0.1 µs after the condensation started under the condition p/ p∗ 1.11 We understand that the liquid film thickness may attain to several nanometers at this time stage. For methanol, on the other hand, the vapor pressure increases up to four to five times the initial pressure at about 0.02 µs after the condensation started under p/ p ∗ 1. The liquid film thickness may attain to several nanometers in this case, too. In short, after a very short time has passed since the condensation began, the liquid films for both

1.280

1.260

0 0

0.2

0.4

p/p *

0.6

0.8

1.0

Fig. 3.13 The relation between refractive index and thickness of the adsorbed liquid film [17]: water

11

The water vapor pressure p at the liquid film surface abruptly increases during the reflection process of the shock wave [see Fig. 3.22b]. During this process, the vapor pressure attains to the saturated vapor pressure p ∗ and the condensation starts at the liquid film surface.

96

3 Methods for the Measurement of Evaporation and Condensation Coefficients

water and methanol develop enough to have their bulk liquid nature in the respective refractive indices. This ensures the validity of the theory of optical interferometer in Sect. 3.4.1. In numerical simulations of the condensation process behind reflected shock waves in Sect. 3.6, initial values of liquid film thickness are needed, and they include a fair uncertainty due to their inexact estimation. However, the uncertainty does not have any influence at all on simulation results, because the thermal process in the liquid film is governed by that of the thick glass; the temperature of liquid film surface changes little because of a large difference in heat capacities between the vapor and the glass. We shall use the initial values of liquid film thickness, estimated from Fig. 3.12, without considering permanently adsorbed liquid films for both methanol and water.

3.6 Deduction of Condensation Coefficient 3.6.1 Typical Output Examples of Energy Reflectance Figure 3.14a, b show typical time changes in the measured energy reflectance of the physical beam for methanol and water vapors; (a) methanol vapor and (b) water vapor. The abscissas are the time for both cases, and the ordinates are the energy reflectance indicated by voltage unit. For methanol vapor, the energy reflectance varies between the two maxima and the two minima as the liquid film grows with time. The maxima are almost the same values, and so are the minima. This means that the liquid film grows uniformly on the glass surface. However, for water vapor, only three quarters of a cycle with a maximum and a minimum are recorded. It is pointed out [11] that the liquid film for water vapor breaks up into small droplets at a certain thickness [11]. The light beam may then be scattered by the droplets and the reflectance may not change regularly anymore. Such a breakup of the liquid film may be due to the fact that the initially adsorbed liquid film is very thin for the

T0 = 295.05 K p0 = 4006.3 Pa Ms = 1.49

Condensation Onset

Time (μs)

(b) Water Voltage (mV)

Voltage (mV)

(a) Methanol

T0 = 292.75 K p0 = 302.64 Pa Ms = 1.93

Condensation Onset

Time (μs)

Fig. 3.14 Typical examples of the time changes of energy reflectance: (a) methanol vapor and (b) water vapor

3.6

Deduction of Condensation Coefficient

97

present initial condition p/ p ∗ ∼ = 0.13, prior to the shock wave reflection. However, it has been demonstrated that the reflectance data for only three quarters of a cycle are enough for the measurement of liquid film thickness [23, 24]. Another difference in the two cases is that the reflectance for water vapor is more disturbed by noises than that for methanol vapor. As a result, it is difficult to decide the instant of condensation onset. The difference may be related to the difference in strength of the incident shock waves; Ms = 1.49 for methanol vapor and Ms = 1.93 for water vapor. Details will be discussed in a later stage of this subsection. The time change in the energy reflectance for Fig. 3.14b is shown in Fig. 3.15a, b, which are both stretched with respect to the time. Figure 3.15a is the reflectance data of both the physical beam (A) and the reference beam (B), and Fig. 3.15b is their difference, (A)–(B). The physical beam (A) is disturbed by systematic noises that appear before the condensation takes place and seriously prevent us from evaluating the thickness of the liquid film. Considering that the photodiode detecting the reference beam (B) is separated from the shock tube, the systematic noises may be generated from the laboratory floor on which the shock tube is fixed, when the diaphragm was burst. The same systematic noises as in (A) appear in the reference beam (B). Figure 3.15b shows that the systematic noises are successfully eliminated by subtracting (B) from (A), and the instant of the condensation onset can be identified. However, we can notice other systematic noises with high frequencies after the condensation onset. According to the frequency analysis of the noises, their frequencies exist in 60–240 kHz (central frequency = 150 kHz), although these values are rough because of the lack of samples of the data. The noises may be caused by vibrations of the glass when the shock wave collided with the glass surface. In fact, a numerical estimation of the natural frequency of the glass embedded in the shock tube endwall is about 143 kHz; this value is obtained by the computation based on finite element method.12 The numerical and experimental natural frequencies are 14

60

50

40 20

(A) Physical Beam

Voltage (mV)

Voltage (mV)

(a)

(B) Reference Beam

25

30 35 40 Time (μs)

45

50

(b)

12 10 8 6 20

25

30 35 40 Time (μs)

45

50

Fig. 3.15 The noise-removed energy reflectance for Fig. 3.14b: (a) physical beam output (A) and reference beam output (B); (b) (A)–(B)

12 The computation was made by Prof. Y. Kobayashi, a specialist of mechanical vibrations at Hokkaido University.

98

3 Methods for the Measurement of Evaporation and Condensation Coefficients 14

Voltage (mV)

Water 12

T0 = 292.75 K

10

p0 = 302.64 Pa Ms = 1.93

8 6 4 20

Condensation Onset

30

40

50

Time (μs)

Fig. 3.16 The instant of condensation onset and the approximated curve in the case of Fig. 3.15b

roughly in agreement. In the following, we will mainly discuss the data of water vapor; the data of methanol vapor corresponding to those of water vapor are given for comparison. Concerning the reflectance for methanol vapor, measurements are conducted by only the physical beam as shown in Fig. 3.14a. Figure 3.16 shows the instant of condensation onset and the approximated curve of energy reflectance in the case of Fig. 3.15b. The dashed line denotes the energy reflectance just before the condensation onset. Before the onset, the surface of initially adsorbed liquid film remains unchanged. After that, the energy reflectance begins to change almost in proportion to the time during about 8 µs. The variation during this time is well approximated by the solid straight line determined by the least square method. The deviation of the reflectance from the straight line can be interpreted as the noises due to vibrations of the glass, as mentioned above. The instant of condensation onset is defined at the intersection point of the solid and dashed lines.

3.6.2 Time Changes of Liquid Film Thickness For deduction of the liquid film thickness, the energy reflectance K (t) in Eq. (3.64) can be read from Fig. 3.16, while K max and K min can be obtained from Fig. 3.14b; K max and K min are needed to normalize the energy reflectance K (t) in the form of s , and I s are [K (t)− K min ]/(K max − K min ). All other quantities such as a+ , a− , Imax min given, so we can evaluate the time change Δd1 (t) of the liquid film thickness using Eq. (3.65). Figure 3.17 shows the time change of the liquid film thickness deduced from the energy reflectance data of Fig. 3.16. The abscissa is the time from the instant of condensation onset and indicated up to 5 µs according to the theoretical prediction in Sect. 3.2.2, and the ordinate is the variation of liquid film thickness, which is not the absolute value. We can notice the slight systematic noises on the datum, but approximate it as the straight line for 5 µs by taking into account the

Deduction of Condensation Coefficient

Liquid Film Thickness (nm)

3.6

99

Water T0 = 292.75 K p0 = 302.64 Pa Ms = 1.93

Time (μs)

Fig. 3.17 The time change of the liquid film thickness for the energy reflectance shown in Fig. 3.16

theoretical prediction and the vibrations of the glass. Substituting the speed of liquid film growth vw , the slope of the straight line, into the KBC given by Eq. (3.23), the numerical simulation of the Gaussian–BGK Boltzmann equation can be carried out without specifying the evaporation and condensation coefficients in advance. The numerical solution gives σ as a function of the time, and hence the αc -value can be determined from Eq. (3.22) with the use of the αe -value (= 0.99) evaluated by MD simulations [14]. The initial time changes of both energy reflectance and liquid film thickness for methanol vapor, obtained from Fig. 3.14a, are shown in Figs. 3.18 and 3.19, respectively. Data acquisition time is chosen to be 0.5 µs according to the theoretical prediction in Sect. 3.2.2. The time change of the liquid film thickness is very similar to the case of water vapor.

Voltage (mV)

Methanol T0 = 295.05 K p0 = 4006.3 Pa

Ms = 1.49

Condensation Onset

Time (μs)

Fig. 3.18 The instant of condensation onset and the approximated curve in the case of Fig. 3.14a

3 Methods for the Measurement of Evaporation and Condensation Coefficients

Liquid Film Thickness (nm)

100

Methanol T0 = 295.05 K p0 = 4006.3 Pa Ms = 1.49

Time (μs)

Fig. 3.19 The time change of the liquid film thickness for the energy reflectance shown in Fig. 3.18

3.6.3 Propagation Process of Shock Waves The numerical solution of the initial and boundary-value problem of the Gaussian– BGK Boltzmann equation, formulated in Sect. 3.2, is given in the present and next subsections. Figure 3.20 shows the simulation results of propagation process of the shock wave advancing toward and reflecting from the shock tube endwall in water vapor under the condition given in Fig. 3.14b; (a) pressure profiles and (b) temperature profiles. The abscissas of (a) and (b) are both the distance from the liquid film surface in the vapor; the distance X is normalized by the mean free path 0 (= 20.75 µm). The time √ indicated for each profile in (a) and (b) is normalized by the mean free time 0 / 2RT0 (= 39.93 ns). The pressure at the interface abruptly rises during the incidence and reflection processes of the shock wave and it reaches

(a) Water 20

30

40

50

60

10

T /T0

p/p0

10

(b) Water

0 –10

20

30

40

50

60

0 –10

–20

–20

X/0

X/0

Fig. 3.20 The propagation process of the shock waves advancing toward and reflecting from the shock tube endwall in water vapor: (a) pressure profiles and (b) temperature profiles. The distance X is normalized by the mean free path 0 (= √20.75 µm), and the numbers near the curves are the times normalized by the mean free time 0 / 2RT0 (= 39.93 ns)

3.6

Deduction of Condensation Coefficient

101 (b) Methanol

10

20

40

30

50

0 –10

–20

60

T /T0

p/p0

(a) Methanol

10

20

30

40

50

60

0 –10

–30

X/0

–20

–30

X/0

Fig. 3.21 The propagation process of the shock waves advancing toward and reflecting from the shock tube endwall in methanol vapor: (a) pressure profiles and (b) temperature profiles. The distance X is normalized by the mean free path 0 (= √ 1.48 µm), and the numbers near the curves are the times normalized by the mean free time 0 / 2RT0 (= 3.79 ns)

the constant value. On the other hand, the temperature profiles show that the thermal boundary layer including the Knudsen layer, i.e., the nonequilibrium region of vapor near the liquid film surface, is formed behind the reflected shock wave. The propagation process of the shock wave in methanol vapor under the condition given in Fig. 3.14a is shown in Fig. 3.21; the mean free path is 1.48 µm, and the mean free time is 3.79 ns.

3.6.4 Time Changes of Macroscopic Quantities and Condensation Coefficient Figure 3.22 shows the simulation results of macroscopic quantities for water vapor at the interface under the condition given in Fig. 3.14b; (a) the vapor temperature T and liquid temperature Tw at the interface, (b) the vapor pressure p and saturated vapor pressure p∗ at Tw , (c) σ and the saturated vapor density ρ ∗ at Tw , and (d) the condensation coefficient αc . At about 0.1 µs after the condensation onset (the origin of the abscissa in Fig. 3.17), the vapor sets in an almost steady state, where the vapor temperature T , pressure p, and σ are almost constant, respectively, while the liquid temperature Tw at the interface continues to rise gradually, and thereby the saturated vapor pressure p ∗ and saturated vapor density ρ ∗ rise, too. The variation in Tw is about 1% for 5 µs, and that in p ∗ is about 20%, resulting in a decrease of the net mass flux of condensation. This is because the net mass flux is in proportion to the difference between p and p ∗ [see Eq. (2.141)], which is a decreasing function of the time, as shown in Fig. 3.22b. As p approaches p ∗ , the vapor–liquid system approaches its equilibrium state, and the value of condensation coefficient approaches that of evaporation coefficient (αe = 0.99). Considering that the vapor state sets in the almost steady one at the time of 0.1 µs after the condensation onset, the evaluation of αc is started at the time of about 0.5 µs. Concerning the evaporation coefficient αe , which is needed in Eq. (3.22) for the deduction of αc , it is taken to

102

3 Methods for the Measurement of Evaporation and Condensation Coefficients

(a)

(b)

p/p0 p∗/p0

T /T0

Tw/T0

(c)

σ/ρ0

(d) αc

ρ∗/ρ0

Time (μs)

Time (μs)

Fig. 3.22 Macroscopic quantities at the interface and the determined αc for water vapor. The origin of the time is reset at the condensation onset. The black circle at the left end on the αc -curve represents the starting time of the acquisition of αc

be 0.99, the value at 290 K, because its dependence upon the temperature Tw is negligibly small for the change in Tw at about 0.5 µs, as will be shown in the next subsection. The macroscopic quantities for methanol vapor under the condition given in Fig. 3.14a are shown in Fig. 3.23. The evaluation of αc is started at about 0.5 µs.

(a)

(b) p/p0 T /T0 p∗/p0 Tw/T0

(c)

σ/ρ0

(d) αc

ρ∗/ρ0

Time (μs)

Time (μs)

Fig. 3.23 Macroscopic quantities at the interface and the determined αc for methanol vapor. The origin of the time is reset at the condensation onset. The black circle at the left end on the αc -curve represents the starting time of the acquisition of αc

3.6

Deduction of Condensation Coefficient

103

3.6.5 Values of αe and αc for Water and Methanol A number of shock tube experiments and the corresponding numerical simulations of the Gaussian–BGK Boltzmann equation with the KBC [Eq. (3.23)] have been made. Deduced values of the condensation coefficient αc are assembled in Fig. 3.24, where we show the relation√ between the αc -values and the σ/ρ ∗ which √ is the ratio of the colliding mass flux σ RTw /(2π ) to the one-way mass flux ρ ∗ RTw /(2π ); ◦ for water and for methanol. We can see that the αc -values of water approach the αe -value, 0.99, as σ/ρ ∗ does unity, where the black circle (•) plotted at σ/ρ ∗ = 1 denotes the αe -value (= 0.99 at 290 K) of water evaluated in MD simulation by Ishiyama et al. [14]. The solid curve is drawn by the least-square fitting so that it may coincide with the αe -value in the limit to the equilibrium state σ/ρ ∗ = 1. The smooth matching of the asymptotic value of αc at σ/ρ ∗ = 1 with the αe value undoubtedly demonstrates that both the measurement of αc -values and the MD simulation are reasonably performed. The relation between αc -values and σ/ρ ∗ for methanol vapor is shown by white squares ( ) and the solid curve; denotes the αe -value (= 0.86 at 290 K) obtained by the MD simulation [14]. We can notice the smooth matching of the asymptotic value of αc at σ/ρ ∗ = 1 with the αe -value in this case, too. A complete theoretical explanation for the results shown in Fig. 3.24 is not yet available. However, we shall try to give it from the following two viewpoints: (i) The difference between the αe -values of water and methanol, (ii) The σ/ρ ∗ -dependence of the αc -values for these materials.

Condensation Coefficient αc

The αe -values of water, methanol, and argon obtained by MD simulations are given in Fig. 3.11. Let us here express the αe -values against the following reduced liquid temperature [16], as shown in Fig. 3.25:

1.0

0.9 0.8 0.7 0.6 Water Methanol

0 1.0

1.1

1.2

σ/ρ∗

Fig. 3.24 The αc -values versus σ/ρ ∗ for water (◦) and methanol ( ). •: αc = αe = 0.99 at 290 K and σ/ρ ∗ = 1, : αc = αe = 0.86 at 290 K and σ/ρ ∗ = 1. The liquid temperature is at about 290 K in the experiments for both water and methanol

104

3 Methods for the Measurement of Evaporation and Condensation Coefficients 1.0 0.8

αe

0.6 0.4 0.2 0 0.4

Water Methanol Argon 0.5

0.6

0.7 T∗

0.8

0.9

Fig. 3.25 The temperature dependence of the αe -values for water, methanol, and argon. The αe values for water and methanol at 290 K are shown by the symbols • (= 0.99) and (= 0.86), and the value for argon at 85 K is shown by the symbol (= 0.87)

T∗ =

Ttriple + A(TL − Ttriple ) , Tcr

(3.67)

where A is a fitting parameter (A = 4/3 for water and methanol, and A = 1 for argon), Ttriple and Tcr are respectively the triple point and critical temperatures (measured values); Ttriple = 273.2 K for water, Ttriple = 175.5 K for methanol, Ttriple = 83.8 K for argon, Tcr = 647.1 K for water, Tcr = 512.6 K for methanol, Tcr = 150.7 K for argon. Figure 3.25 shows that evaporation coefficients of different species of molecules can be described by a universal function of the reduced temperature T ∗ , although the physical meaning of the definition (3.67) has not been clarified yet. The existence of the universal function of T ∗ suggests that, irrespective of the fact that the molecular motions of water and methanol are affected by Coulombic forces, the temperature dependence of evaporation coefficients may be explained by the same argument as that of argon. The αe -values of water, methanol, and argon quantitatively agree well against T ∗ and decrease with increasing T ∗ and become close to unity with decreasing T ∗ ; at least the αe -values of water and argon are almost unity at T ∗ 0.48. In the case of water, the temperature Tw at T ∗ = 0.48 is 301 K. The αe -values for water and methanol at 290 K are shown by the symbols • (αe = 0.99) and (αe = 0.86), and the αe -value for argon at 85 K near its Ttriple is shown by the symbol (αe = 0.87). The σ/ρ ∗ -dependence of the αc -values shown in Fig. 3.24 may be explained as follows. From Eq. (3.22), we obtain the relation between αc and σ/ρ ∗ : 1 ρ L vw αe + ∗ √ , αc = σ/ρ ∗ ρ RTw /(2π )

(3.68)

which satisfies αc = αe for σ/ρ ∗ = 1 because vw = 0 at the equilibrium state. This can also be obtained from Eq. (2.146). We understand that the condensation

3.6

Deduction of Condensation Coefficient

105

∗ coefficient αc decreases inversely proportional to √ σ/ρ and slightly increases with ∗ increasing ρ L vw . The magnitudes of ρ L vw /[ρ RTw /(2π )] for water, the second term in the square brackets, are 0.029 for σ/ρ ∗ = 1.05 and 0.055 for σ/ρ ∗ = 1.1 (Fig. 3.24). For understanding the reason of the decrease in αc with the increase in σ/ρ ∗ , let us pay attention to the reflection and condensation mass fluxes, Jref and Jcnds , given by Eqs. (3.18) and (3.19). From Eqs. (3.18), (3.19), (3.20), and (3.21), we obtain

Jref ρ L vw σ √ = ∗ − αe − ∗ √ , ρ ρ ∗ RTw /(2π ) ρ RTw /(2π ) ρ∗

√

ρ L vw Jcnds = αe + ∗ √ . RTw /(2π ) ρ RTw /(2π )

(3.69) (3.70)

For σ/ρ ∗ = 1, the right-hand sides of Eqs. (3.69) and (3.70) become 1 − αe and αe respectively, as they should do so. Estimating the values of the normalized reflection and condensation mass fluxes, the former is 0.031 for σ/ρ ∗ = 1.05 and 0.055 for σ/ρ ∗ = 1.1, and the latter is 0.961 for σ/ρ ∗ = 1.05 and 0.935 for σ/ρ ∗ = 1.1. We understand that the reflection mass flux changes by about 2 times for variations from 1.05 to 1.1 in σ/ρ ∗ , while the condensation mass flux changes by 2.7% at the most for the same variations in σ/ρ ∗ . This means that the condensation mass flux is not sensitive at all to the increase in σ/ρ ∗ , in other words, the amount of vapor molecules adsorbed into the liquid at the interface is almost constant, and instead molecules which are not adsorbed reflect at the interface. This can also be understood from the following relation obtained from Eqs. (3.69) and (3.70): Jref ∼ σ − αe ρ ∗ , = Jcnds αe ρ ∗

(3.71)

where αe is treated as O(1), and the last terms in Eqs. (3.69) and (3.70) are neglected because they are of the order of k; k is defined in Eq. (2.107). For water vapor, the numerator in the right-hand side of Eq. (3.71) is almost the difference between σ and ρ ∗ because αe is nearly unity (αe = 0.99), and thus Jref /Jcnds is nearly equal to σ/ρ ∗ − 1. This means that the excess mass per unit volume of vapor molecules against the interface over the saturated density is reflected at the interface. The density ratio σ/ρ ∗ is given by Eq. (2.146). Equation (2.146) can also be used for evaluating σ/ρ ∗ in Fig. 3.24, when we use the αc -values in fluid-dynamic equations (2.138) and (2.139). Returning to the σ/ρ ∗ -dependence of the αc values shown in Fig. 3.24, from Eqs. (3.68) and (3.69), we obtain αc = 1 −

Jref 1 . √ ∗ ∗ σ/ρ ρ RTw /(2π )

(3.72)

Equation (3.72) represents that the condensation coefficient αc is governed by the product of the reciprocal of ρ ∗ /σ and the normalized reflection mass flux

106

3 Methods for the Measurement of Evaporation and Condensation Coefficients

√ Jref /[ρ ∗ RTw /(2π )]. As discussed in the above, the ratio σ/ρ ∗ for water vapor changes in the range of 1–1.2 in Fig. 3.24, and on the other hand, the normalized reflection mass flux changes in the range of 0–0.12. The αc -values correspondingly changes from 0.99 at the equilibrium state to about 0.9 as the ratio σ/ρ ∗ increases. In the above discussion, we focused on the σ/ρ ∗ -dependence of αc when the net condensation occurred in the nonequilibrium states. We should give a comment on the σ/ρ ∗ -dependence of αc when net evaporation takes place in a nonequilibrium state. In this case, it is expected that the ratio σ/ρ ∗ is slightly smaller than unity, but almost unity because the vapor is in an almost equilibrium state at the interface; the evaporation is not the virtual vacuum evaporation. Considering that the second term in the square brackets is negative for the evaporation, we may expect that the αc value is approximately equal to the αe -value, as the case of the weak condensation discussed in the above. This is the limit for what we can discuss about the αc values. It would be a challenging work to investigate further detailed mechanisms of the interaction of molecules such as reflection mechanism of molecules at the interface.

3.7 Sound Resonance Method for Measurement of Evaporation Coefficient In Sect. 3.6, it has been demonstrated that the values of condensation coefficient for water and methanol approach their respective values of the evaporation coefficient, evaluated by the MD simulations, as the vapor states approach their equilibrium ones. This result implies that the values of the evaporation coefficient αe are reasonable ones. However, these are not directly measured values. Any direct measurements of αe are needed in order to thoroughly resolve the problem of the evaporation and condensation coefficients. To measure the evaporation coefficient directly, we have two options: (i) To measure a vapor state accompanied with very weak evaporation or condensation, a vapor state i.e., very close to an equilibrium one, (ii) To measure the evaporation mass flux Jevap in a condition that the mass flux at the interface is negligibly small. Since the second option (ii) is very difficult, we adopt the first one (i) and disregard the difference of the vapor state from the equilibrium.13 In this section, a new method is introduced for the measurement of the evaporation coefficient for the very weak evaporation and condensation induced by sound waves in the vapor, where a typical Mach number is 0.001 at the most,

13

The flux must be measured in a nonequilibrium condition. However, as discussed in Sect. 1.2, most of the previous experimental works on the evaporation coefficient have encountered the difficulty in realization of experiments in the scale of the transition time τt defined by Eq. (1.21).

3.7

Sound Resonance Method for Measurement of Evaporation Coefficient

107

although it is now under development in the authors’ laboratories. This method is also promising for investigation of nonlinear nonequilibrium phenomena in acoustical physics [3, 12, 28]. Variations of the pressure, velocity, and temperature of the vapor, induced by the sound waves, are sufficiently small compared with those in a reference state. Very weak nonequilibrium states can be realized in the neighborhood of a vapor–liquid interface. In such states, the evaporation and condensation coefficients can be regarded as nearly identical, i.e., α = αe = αc . As shown in Fig. 3.26, we consider the system, the space of which is bounded by a sample liquid and a circular sound source, and filled with the saturated vapor of the liquid. The plane surface of sound source is parallel to the liquid surface, i.e. the plane vapor–liquid interface, and the distance between the sound source and the interface is about one wavelength of the sound in the vapor at a driving frequency of the sound source. A diameter of the sound source is large enough compared to the wavelength so that the assumption of plane waves may be satisfied with a high accuracy. The system including the sound source is insulated by a circular tube with almost the same diameter as that of the sound source. Because of this covering, the formation of plane sound waves is expected. In this system, the sound resonance is excited when the distance is equal to integral multiples of half wavelength. Two receivers are installed in order to detect the variation in pressure of a test vapor in the space; Receiver 1 is attached to the surface of sound source and Receiver 2 is placed under the sample liquid. The method of measurement consists of an experimental part and a theoretical one. In the experimental part, we obtain the amplitude of vapor pressure under the sound resonance induced by the sound waves. Then, the pressure amplitude is substituted into a relation between the evaporation coefficient and the pressure amplitude, obtained from the theoretical part, and thereby the evaporation coefficient is deduced. Details of the experimental and theoretical parts are as follows. First, the sound source starts the continuous oscillation at a constant frequency and a constant displacement. Then, the plane sound wave propagates in the vapor space and after a few cycles, a standing wave is formed. When the distance between the sound source and the interface is equal to one wavelength, the sound resonance of the second mode is excited and the nonlinear effect arises with the increase in the wave amplitude. Figure 3.27 shows the waveform detected by Receiver 2 placed on a solid wall in the case that the distance L is one wavelength and the initial pressure is 101 kPa

Sound Source

L

Receiver 1 Standing Wave Liquid Receiver 2

Fig. 3.26 The schematic of system

108

3 Methods for the Measurement of Evaporation and Condensation Coefficients

OutputVoltage (V)

0.015 0.010 0.005 0 –0.005 0

2

4 6 Time (μs)

8

10

Fig. 3.27 The wave profile at the sound resonance (solid wall, 101 kPa)

(atmospheric pressure). At the sound resonance, the amplitude of the standing wave increases without increase in input power of the sound source and the effect of evaporation and condensation may be amplified. Very weak evaporation and condensation take place cyclically, and a certain amount of the sound wave incident on the interface is absorbed into the liquid. The amplitude of the standing wave is thus affected by the evaporation and condensation. In this experimental method, we measure the output signal at a resonant point (sound resonance) by Receiver 1 and Receiver 2, and retrieve the output voltage amplitude of the second harmonics through the frequency analysis. The second harmonics is utilized to enhance the signal-to-noise ratio, because the output signal contains electromagnetic noises of the driving frequency. As mentioned at the beginning of this section, the method is under development with a great prospect in the authors’ laboratories [28].

References 1. P. Andries, P.L. Tallec, J.P. Perlat, B. Perthame, The Gaussian-BGK model of Boltzmann equation with small Prandtl number. Eur. J. Mech. B- Fluids 19, 813–830 (2000) 2. H. S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, 2nd edn. (Oxford University Press, Oxford, 1959) 3. W. Chester, Resonant oscillations in closed tubes. J. Fluid Mech. 18, 44–64 (1964) 4. A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Tables of Integral Transforms, vol. I (McGraw-Hill, New York, NY, 1954) 5. S. Fujikawa, M. Okuda, T. Akamatsu, T. Goto, Non-equilibrium vapour condensation on a shock-tube endwall behind a reflected shock wave. J. Fluid Mech. 183, 293–324 (1987) 6. S. Fujikawa, M. Kotani, H. Sato, M. Matsumoto, Molecular study of evaporation and condensation of an associating liquid: Shock-tube experiment and molecular dynamics simulation. Heat Transfer-Japan. Res. 23, 595–610 (1994) 7. S. Fujikawa, M. Kotani, N. Takasugi, Theory of film condensation on shock-tube endwall behind reflected shock wave: Theoretical basis for determination of condensation coefficient. JSME Int. J. 40, 159–165 (1997) 8. S. Fujikawa, Y. Takano, T. Akamatsu, T. Mizutani, Computational simulation of film condensation of carbon tetrachloride vapor on the end- and the sidewall of a shocktube, in Proceedings of IUTAM Symposium Aerothermochem. Spacecraft and Associated

References

9.

10. 11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22. 23. 24. 25.

26. 27. 28. 29. 30.

109

Hypersonic Flows, eds. by R. Brun, A.A. Chikhaoui (Le Palais du Pharo, Marseille, 1992), pp. 95–100 S. Fujikawa, T. Yano, K. Kobayashi, K. Iwanami, M. Ichijo, Molecular gas dynamics applied to phase change processes at a vapor-liquid interface: Shock-tube experiment and MGD computation for methanol. Exp. Fluids 37, pp. 80–86 (2004) A.G. Gaydon, I.R. Hurle, The Shock Tube in High-Temperature Chemical Physics (Chapman and Hall, London, 1963) R. Goldstein, Study of water vapor condensation on shock-tube walls. J. Chem. Phys. 40, 2793–2799 (1964) M. Inaba, S. Fujikawa, T. Yano, Molecular gas dynamics on condensation and evaporation of water induced by sound waves. Rarefied Gas Dynamics. AIP Conference Proceedings, vol. 1084 (AIP, Melville, NY, 2009), pp. 671–676 T. Ishiyama, T. Yano, S. Fujikawa, Molecular dynamics study of kinetic boundary condition at an interface between argon vapor and its condensed phase. Phys. Fluids 16, 2899–2906 (2004) T. Ishiyama, T. Yano, S. Fujikawa, Molecular dynamics study of kinetic boundary condition at an interface between polyatomic vapor and its condensed phase. Phys. Fluids 16, 4713–4726 (2004) T. Ishiyama, T. Yano, S. Fujikawa, Kinetic boundary condition at a vapor-liquid interface. Phys. Rev. Lett. 95, 084504 (2005) T. Ishiyama, T. Yano, S. Fujikawa, Molecular dynamics study on the evaporation part of the kinetic boundary condition at the interface between water and water vapor. Rarefied Gas Dynamics. AIP Conference Proceedings, vol. 762 (AIP, Melville, NY, 2005), pp. 491–496 K. Kinosita, H. Kojima, H. Yokota, Adsorption of water vapour on a cleavage surface of lithium fluoride. Japan. J. Appl. Phys. 1, 234 (1962) K. Kobayashi, A Study on Condensation Coefficients of Methanol and Water in Weak Condensation States Based on Molecular Gas Dynamics (Ph.D Thesis, Hokkaido University, Sapporo, 2007) (in Japanese) K. Kobayashi, S. Watanabe, D. Yamano, T. Yano, S. Fujikawa, Condensation coefficient of water in a weak condensation state. Fluid Dyn. Res. 40, 585–596 (2008) M. Kotani, T. Tsuzuyama, Y. Fujii, S. Fujikawa, Molecular study on nonequilibrium condensation of acetic acid vapor, in Microscale Heat Transfer, eds. by J.B. Saulnier, D. Lemonnier, J.P. Bardon (Laboratoire d’ Etudes Thermiques, Poitiers, 1998), pp. 95–100 J.D. Lambert, G.A.H. Roberts, J.S. Rowlinson, V.J. Wilkinson, The second virial coefficients of organic vapours. Proc. R. Soc. Lond. A 196, 113–125 (1949) J.E. Lowder, Increase of integrated intensities of H2 O infrared bands produced by hydrogen bonding. J. Quant. Spectrosc. Radiat. Transfer 11, 153–159 (1971) M. Maerefat, Studies on Nonequilibrium Vapor Condensation by Shock Tube (Ph.D Thesis, Kyoto University, Kyoto, 1990) M. Maerefat, T. Akamatsu, S. Fujikawa, Non-equilibrium condensation of water and carbontetrachloride vapour in a shock-tube. Exp. Fluids 9, 345–351 (1990) M. Maerefat, S. Fujikawa, T. Akamatsu, Non-equilibrium condensation of a vapour-gas mixture on a shock-tube endwall behind a reflected shock wave. Fluid Dyn. Res. 6, 25–42 (1990) M. Maerefat, S. Fujikawa, T. Akamatsu, T. Goto, T. Mizutani, An experimental study of nonequilibrium vapour condensation in a shock-tube. Exp. Fluids 7, 513–520 (1989) P.M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, NY, 1953) S. Nakamura, T. Yano, M. Watanabe, S. Fujikawa, The sound wave method for measurement of evaporation coefficient. J. Fluid Sci. Technol. 5, 26–34 (2010) S. Schiller, U. Heisig, Bedampfungs-technik: Verfahren Einrichtungen Anwendungen (VEB, Berlin, 1975) W.G. Vincenti, C.H. Kruger, Jr., Introduction to Physical Gas Dynamics (Krieger, Florida, 2002)

Chapter 4

Vapor Pressure, Surface Tension, and Evaporation Coefficient for Nanodroplets

Abstract For very small bubbles or droplets, the vapor pressure, surface tension, evaporation coefficient, and relations among them at the vapor–liquid interface are greatly influenced by not only the liquid temperature but also the curvature of the interface. In this chapter, the molecular dynamics simulation of equilibrium states of argon nanodroplets clarifies the validity and limitation of the Laplace equation and the Kelvin equation. The Laplace equation is derived in the purely mechanical way and is proved to hold in nanoscale, however, the Kelvin equation, the thermodynamics equation for the vapor–liquid system, does not hold for extremely small nanodroplets with 1.5 nm in radius. The values of surface tension decrease with increasing temperature and quantitatively agree well with the experimental values in the temperature range investigated. The nonequilibrium MD simulation is also made to evaluate the evaporation coefficient at the interface of argon nanodroplets. We find the control surface suitable for counting molecules across it, where the mass flux of evaporating molecules from a nanodroplet into the vacuum is equal to that from the planar liquid film. With the use of the control surface, the evaporation coefficient of nanodroplets is accurately evaluated and the theoretical formula describing the dependence of evaporation coefficient on the droplet size from a few nanometers to infinity is obtained.

4.1 Significance of Molecular Dynamics Analysis for Nanodroplets A single-component vapor–liquid two-phase system consisting of a liquid droplet and its surrounding vapor in an equilibrium state is a fundamental system important in various engineering applications; spray flow, droplet combustion, and microparticle formation. This system has several well-known characteristics, such as the increase of the vapor pressure surrounding the droplet with decreasing droplet size at a fixed temperature, the droplet radius dependence of surface tension in addition to the temperature dependence, and so on. These characteristics can be theoretically explained, e.g., by the Kelvin equation [9, 14] for the former and by the Tolman equation [15] for the latter. The derivation of the Kelvin and Tolman equations is outlined in Sects. 4.4.6 and 4.4.7. These equations are, however, on the basis of

S. Fujikawa et al., Vapor-Liquid Interfaces, Bubbles and Droplets, Heat and Mass C Springer-Verlag Berlin Heidelberg 2011 Transfer, DOI 10.1007/978-3-642-18038-5_4,

111

112

4 Vapor Pressure, Surface Tension, and Evaporation Coefficient

thermodynamics which assumes an equilibrium state consisting of a great number of molecules. Here, a question is raised whether thermodynamics is valid or not for a small droplet consisting of one thousand molecules at the most; hereafter, such a small droplet will be called nanodroplet. The applicability of thermodynamics to phenomena of nanoscale in length is of fundamental importance in fluid mechanics based on the continuum hypothesis, because the local equilibrium is an essential prerequisite for the continuum hypothesis. Therefore, we carefully investigate this issue with molecular dynamics (MD) simulations of the equilibrium states of single argon nanodroplets and their surrounding argon vapor. A number of MD studies have been conducted for equilibrium states of nanodroplets surrounded by their vapor. Thompson et al. [13] systematically examined the droplet-vapor system with a relatively large number of molecules in spite of a limited amount of computer resources available in 1980s. A still more exhaustive MD study has been executed by Vrabec et al. [16]. However, the applicability of thermodynamics to nanodroplets has never been addressed, and all previous studies have just confirmed the qualitative agreement between thermodynamical results, such as the Kelvin and Tolman equations, and the MD results for nanodroplets of radii larger than a few nanometers [8]. Although Thompson et al. [13] have mentioned the possibility of breakdown of the Kelvin equation for nanodroplets of radii less than a few nanometers, the question on the applicabitily of thermodynamics to nanodroplets still remains unresolved. In this chapter, based on MD simulations, we discuss the applicability of thermodynamics to equilibrium states of single argon nanodroplets and their surrounding vapor from microscopic information of molecular motions [17]. For such a critical discussion, the spatial distributions of macroscopic variables, especially pressure, should be evaluated precisely. To this end, we evaluate the whole momentum transported across an arbitrary unit area per unit time, thereby giving precise normal and tangential pressures including their local variations. The surface tension is then given by the mechanical balance for a small volume element. The MD simulations are performed for more than 30 cases of various droplet radii ranging from about 1 to 4 nm and system temperatures from 85 to 105 K. It should be noted that the triple point temperature of argon is 83.8 K and the critical temperature is 150.7 K. The system temperatures chosen here are sufficiently low compared with the critical temperature so that the vapor may be regarded as an ideal gas. Considering the qualitative difference between the Laplace equation derived from the mechanical equilibrium condition and the Kelvin equation from thermodynamical equilibrium condition, we obtain the following conclusions; (A) the minimum radius with which a single nanodroplet can stably exist for a rather long time over 200 ns is about 1 nm and it is 0.3 nm at 85 K, (B) the Laplace equation holds for all single nanodroplets with about 1 to 4 nm in radius at the temperatures between 85 and 105 K, (C) although a nanodroplet of radius about 1 nm can exist stably in the equilibrium state, the balance of chemical potentials for the bulk liquid and bulk vapor does not hold for such a small droplet, and consequently the Kelvin equation based on their balance does not hold.

4.2

Method of MD Simulations

113

4.2 Method of MD Simulations Let N , V , and E be the total number of molecules in the system to be considered, the total volume, and the total energy, respectively. MD simulations are carried out under the condition with N , V , and E kept constant (i.e., N V E simulation). Temperatures of the system are chosen as 85, 90, 95, 100, and 105 K, and radii of nanodroplets range from 1 to 4 nm; the radius will be defined correctly in a later stage. MD simulations are also performed for the planar interfaces of liquid films at these temperatures, which correspond to the nanodroplets of infinite radius. Translational motion of argon molecules obeys Newton’s second law of motion, and an intermolecular potential is given by the Lennard-Jones 12-6 potential: φ(ri j ) = 4

σ ri j

12

σ − ri j

6 ,

(4.1)

where ri j is the distance between the centers of a molecule i and a molecule j, and it is defined as ri j = |r i − r j | in which r i is the position vector of the molecule i. Given the intermolecular potential (4.1), the magnitude of the force acting between the molecules i and j is expressed by F i j = −dφ/dri j , where a positive value of F i j denotes repulsion and a negative one denotes attraction. The symbols and σ are potential parameters; /k = 119.8 K (k is the Boltzmann constant) and σ = 0.341 nm for argon. The cut-off radius rc at which the influence of intermolecular forces among molecules actually disappears is set to be 1.5 nm (= 4.4 σ ). Results with this cut-off radius are not much different from those with rc = ∞ [7]. The equations of motion are integrated by the leap-frog method with the time step of 5 fs. The MD simulations of the vapor–nanodroplet system are performed in a cubic simulation cell with the volume of L × L × L, and those of the vapor–planar liquid film system are done in a rectangular cell with the volume of L x × L y × L z , as shown in Fig. 4.1. The periodic boundary condition is adopted for all three directions of the simulation cells. The cell length L is chosen to be several times radius of each nanodroplet. It should be mentioned that the above parameters such as the cut-off radius, time step, and cell length etc. are determined properly, so that the total translational momentum, density distribution, and droplet radius are not affected by these parameters. Although the total translational momentum of the system is always zero, the gravity center of the system fluctuates in time because the periodic boundary condition is used. Spatial configurations of all molecules are therefore to be shifted so that the gravity centers of both nanodroplet and liquid film may coincide with the geometrical centers of the cells; this procedure does not affect simulation results at all. Special attention must be paid for the preparation of a vapor–liquid equilibrium state, as described below. First, the initial configuration of vapor and liquid molecules is appropriately set to be adaptable to the temperature and droplet radius determined. Computation is performed until the system reaches its equilibrium state by the temperature control based on the velocity scaling method for the time lapse

114

4 Vapor Pressure, Surface Tension, and Evaporation Coefficient

Nanodroplet

Interface

Liquid Film

Interface

(a)

(b)

Fig. 4.1 (a) The nanodroplet and (b) the planar liquid film in vapor–liquid equilibrium states at 85 K

of more than 100 ns. Then, a preparatory computation is carried out without the temperature control for the time lapse of more than 200 ns until the realization of the equilibrium state is confirmed. The criterion for the realization of the equilibrium state is that macroscopic quantities such as the temperature and droplet radius become steady and the velocity distribution becomes the Maxwellian everywhere in the stationary system. After the careful confirmation of the equilibrium state, the configuration in the system is adopted as the initial state and the main computation is performed during 201.6 ns without the temperature control (N V E simulation) in order to obtain macroscopic physical quantities. Figure 4.1a shows the argon nanodroplet and its surrounding vapor in the equilibrium state at 85 K. Macroscopic physical quantities such as density, pressure etc. in the spherical coordinate system (r, θ, φ) are obtained as the time-averaged values for the duration of 201.6 ns; the origin of the system coincides with the gravity center of the nanodroplet. Time-averaged spatial distributions of macroscopic physical quantities display spherical symmetry each; hence macroscopic physical quantities are averaged with respect to θ and φ as well as the time t, and then these are rewritten as functions of the radial direction r only. Figure 4.1b shows the planar liquid film of argon and the surrounding vapor in the equilibrium state at 85 K. The computational conditions for the liquid film are similar to those for the nanodroplet. The total number of molecules is N = 4000, and the temperatures are set at 85, 90, 95, 100, and 105 K. Cartesian coordinates (x, y, z), the origin of which coincides with the gravity center of the liquid film, are adopted. The coordinate z stems from the interface outward. The computation cell is the rectangular parallelepiped with L x = L y = 5 nm and L z = 30 nm. Macroscopic physical quantities are averaged with respect to x and y as well as the time t, and z-distributions of these quantities are obtained. MD simulations and data processing methods by Ishiyama et al. [2, 4] are used in the present MD simulations.

4.3

Computational Method of Pressures

115

4.3 Computational Method of Pressures Generally, pressures acting on a surface are defined as the sum of both the momentum given by molecules passing through unit area of the surface per unit time and the intermolecular force acting from one side to the other one across the unit area. According to this pure mechanical definition, a pressure pk on the surface is given by + , 1 i 1 i j |(r i − r j ) · nk | pk = mξ · nk + F , At A |r i − r j | i,t

ij

(4.2)

M

where A is the area of a control surface considered, nk is the unit vector normal to the surface (k indicates the direction of the unit normal vector), m is the mass of one molecule, ξ i is the velocity of a molecule i passing through the control surface, F i j is the intermolecular force between the molecule i and a molecule j interacting with each other, between which the control surface exists [Fig. 4.2b]. The summation i,t of the first term in the right-hand side of Eq. (4.2) means the one for all molecules passing through the control surface during the time duration t [see Fig. 4.2a]. The passage of all molecules through the control surface is counted every time - step of 5 fs during t = 201.6 ns, the whole computation time. The summation i j of the second term means the sum of intermolecular force F i j among molecules across the control surface at an arbitrary time [see Fig. 4.2b]. The angle brackets M denote the average of M sums at different times. We get samples of the above quantities every 10 time steps and put M = 4.032 × 106 for nanodroplets larger than about 1.7 nm in radius, while we get samples every time step and put M = 4.032 × 107 for nanodroplets smaller than it. The pressure evaluated by Eq. (4.2) gives the pressure pr , the pressure acting on the surface of r = const. by taking nk in the r -direction, the pressure pθ , the pressure acting on the surface of θ = const. by taking nk in the θ -direction, and the pressure pφ , the pressure acting on the surface of φ = const. by taking nk

i

ξi

F ij

j

j nk

(a)

nk

ri

rj

(b)

Fig. 4.2 Schematics of pressure calculation: (a) contribution of molecular motion and (b) that of intermolecular force

116

4 Vapor Pressure, Surface Tension, and Evaporation Coefficient

in the φ-direction. It should be noted that the magnitudes of these pressures near the interface depend on the directions of control surfaces unlike the pressure in a homogeneous and stationary fluid.

4.4 Equilibrium States of Nanodroplets and Planar Liquid Films 4.4.1 General Explanation Parameters and physical quantities for nanodroplets and planar liquid films at their equilibrium states are shown in Tables 4.1 and 4.2, where ρV is the vapor density, R is the gas constant per unit mass, pV is the vapor pressure surrounding nanodroplets, p∞ is the saturated vapor pressure for liquid films, and the definitions and explanation of Rs , Rm , δ, ρ L , and γ will be given later; R0√and Δ are explained in Sect. 4.4.7. For reference, the mean free path = m/( 2π σ 2 ρV ) of vapor molecules [11] and the compression factors, i.e., pV /(ρV RT ) for the nanodroplets and p∞ /(ρV RT ) for the liquid films, are also shown in Tables 4.1 and 4.2. Figures 4.3, 4.4, 4.5, and 4.6 show the r -distributions of density and pressure for the nanodroplets, and Figs. 4.7 and 4.8 show the z-distributions of density and pressure for the liquid films. Physical quantities at an arbitrary position r or the z-direction are obtained by averaging their relevant quantities with respect to the time within a thin shell at r for the nanodroplets or a thin layer at z for the liquid films. It was found that the vapor–liquid equilibrium, which is stably sustained for the duration of 201.6 ns, and also at which the surface tension is able to be well defined, was not obtained for nanodroplets smaller than those listed on Table 4.1, although computations were carried out with a wide variety of choices of parameters, such as molecular number N , time interval of temperature control, computation cell size, initial configuration, and so on. This result strongly suggests that the minimum radii of nanodroplets at each temperature shown in Table 4.1 are the approximate values of the minimum radii with which the nanodroplets can stably exist. It should be noted that there exist no other studies on the minimum radii with which the nanodroplets can stably exist for the long time over 200 ns at different temperatures.

4.4.2 Density Distributions In density distributions shown in Figs. 4.3, 4.4, 4.5, 4.6, 4.7, and 4.8, the high density regions of the left-hand side are the liquid phase whose density is ρ L , while the low density regions are the vapor phase whose density is ρV . The regions with abrupt changes in both density and pressure are called the transition layers. The position and thickness of the transition layer in density are defined by the central position Rm

N

515 700 515 500 515 450 515 730 1000 2000 4000 8000

515 1737 730 1000 2000 4000 8000

(K)

85 85 85 85 85 85 85 85 85 85 85 85

90 90 90 90 90 90 90

T

9.5 18.0 10.0 10.5 13.0 16.5 21.0

11.5 13.0 12.0 12.0 12.0 11.0 9.5 10.0 10.5 13.0 16.5 21.0

(nm)

0.98 1.08 1.32 1.58 2.17 2.89 3.78

0.27 0.41 0.51 0.63 0.74 0.84 1.16 1.43 1.70 2.27 2.99 3.91

1.46 1.53 1.76 2.03 2.62 3.34 4.23

0.86 0.88 0.97 1.06 1.18 1.26 1.58 1.84 2.09 2.68 3.40 4.31 1.51 1.58 1.81 2.07 2.65 3.36 4.24

0.94 0.96 1.03 1.11 1.23 1.31 1.61 1.88 2.12 2.70 3.42 4.32 0.53 0.50 0.48 0.48 0.48 0.46 0.47

0.67 0.55 0.52 0.48 0.49 0.47 0.46 0.45 0.42 0.43 0.43 0.40 0.74 0.74 0.73 0.77 0.78 0.78 0.79

0.66 0.71 0.65 0.65 0.66 0.66 0.68 0.67 0.72 0.71 0.71 0.72 7.7 8.1 8.9 9.7 11.0 12.0 12.8

6.6 6.8 8.0 8.8 9.5 10.3 12.0 13.9 14.9 17.0 18.9 20.6 1401 1399 1397 1395 1392 1385 1380

1379 1371 1407 1414 1421 1430 1428 1429 1427 1420 1416 1412

(kg/m3 )

16.61 15.87 14.36 13.26 11.65 10.65 10.02

19.33 18.83 16.07 14.55 13.42 12.46 10.72 9.19 8.58 7.55 6.77 6.21 16.2 15.4 13.5 12.4 10.1 7.9 6.3

11.9 12.3 15.9 16.7 18.8 17.7 16.2 14.8 12.8 10.6 8.4 6.6

(MPa)

0.283 0.273 0.251 0.234 0.205 0.190 0.178

0.303 0.298 0.254 0.234 0.219 0.205 0.177 0.154 0.144 0.126 0.115 0.106

0.0078 0.0081 0.0088 0.0096 0.0107 0.0112 0.0116

0.0016 0.0025 0.0040 0.0052 0.0069 0.0073 0.0093 0.0104 0.0107 0.0119 0.0124 0.0127

(N/m)

Table 4.1 The parameters and numerical results of vapor–liquid equilibrium states of argon nanodroplets and their surrounding vapor L Rs Rm R0 Δ δ ρL ρV pL pV γ

0.91 0.92 0.93 0.94 0.94 0.95 0.95

pV ρV RT 0.89 0.90 0.90 0.91 0.92 0.93 0.93 0.95 0.95 0.95 0.96 0.96

4.4 Equilibrium States of Nanodroplets and Planar Liquid Films 117

(nm)

N

515 730 1000 2000 4000 8000

730 1000 2000 4000 8000

1500 2000 4000 8000

(K)

95 95 95 95 95 95

100 100 100 100 100

105 105 105 105

12.0 13.0 16.5 21.0

10.0 10.5 13.0 16.5 21.0

9.5 10.0 10.5 13.0 16.5 21.0

L

T

0.96 1.53 2.25 2.99

0.80 1.23 1.87 2.55 3.39

0.66 1.11 1.45 2.09 2.77 3.67

Rs

1.59 2.09 2.75 3.58

1.33 1.75 2.38 3.06 3.90

1.17 1.63 1.93 2.52 3.22 4.10

Rm

1.68 2.15 2.80 3.61

1.42 1.81 2.42 3.09 3.93

1.24 1.68 1.98 2.56 3.25 4.12

R0

0.72 0.62 0.55 0.63

0.62 0.58 0.55 0.55 0.54

0.59 0.58 0.53 0.47 0.48 0.45

Δ

0.98 0.97 0.98 0.98

0.89 0.87 0.88 0.89 0.91

0.78 0.81 0.80 0.87 0.86 0.86

3.0 3.5 3.8 4.2

3.9 4.4 5.1 5.5 5.9

4.7 5.9 6.5 7.3 7.8 8.4

Table 4.1 (continued) δ

1289 1297 1290 1287

1333 1331 1330 1322 1317

1354 1370 1366 1358 1352 1348

(kg/m3 )

ρL

43.30 36.51 33.49 30.49

32.97 29.27 25.17 23.21 21.78

27.35 21.64 19.62 17.55 16.38 15.24

ρV

10.2 8.7 6.7 5.6

10.8 10.7 8.8 7.2 5.7

12.5 13.6 11.8 8.9 7.3 5.7

(MPa)

pL

0.793 0.684 0.636 0.584

0.585 0.534 0.468 0.435 0.411

0.467 0.385 0.350 0.319 0.300 0.280

pV

0.0045 0.0061 0.0068 0.0075

0.0041 0.0062 0.0078 0.0086 0.0090

0.0040 0.0073 0.0083 0.0089 0.0097 0.0099

(N/m)

γ

0.84 0.86 0.87 0.88

0.85 0.88 0.89 0.90 0.91

pV ρV RT 0.86 0.90 0.90 0.92 0.93 0.93

118 4 Vapor Pressure, Surface Tension, and Evaporation Coefficient

4.4

Equilibrium States of Nanodroplets and Planar Liquid Films

119

Table 4.2 The parameters and numerical results of vapor–liquid equilibrium states of argon planar liquid films and their surrounding vapor. The values in parentheses are experimental ones T

Zs

(K)

(nm)

85 90 95 100 105

3.44 3.47 3.51 3.55 3.56

Zm

Δ

Z0

δ

ρL

ρV

(kg/m3 ) 3.78 3.85 3.90 3.96 4.00

3.77 3.83 3.89 3.94 3.99

0.33 0.36 0.38 0.39 0.42

p∞

γ∞

(MPa)

(N/m)

0.64 26.9 1394 4.76 0.083 (0.079) 0.69 16.4 1362 7.81 0.141 (0.133) 0.76 10.6 1329 12.03 0.225 (0.213) 0.82 7.4 1297 17.40 0.338 (0.324) 0.89 5.2 1263 24.47 0.484 (0.472)

0.0136 0.0124 0.0111 0.0100 0.0088

(0.0131) (0.0118) (0.0106) (0.0094) (0.0082)

p∞ ρ V RT 0.98 0.96 0.95 0.93 0.90

Density ρ (kg/m3)

and 10-90 thickness δ of the transition layer according to Refs. [2, 4] and these are shown in Tables 4.1 and 4.2, where R0 is the position of the Gibbs dividing surface (equimolor dividing surface) for the nanodroplets obtained from Eq. (4.19) and Z 0 is that for the liquid films obtained from Eq. (4.20); Eqs. (4.19) and (4.20) will be derived in Sect. 4.4.7.

(a)

pr Pressure p (MPa)

pθ pφ

Eq. (4.3) (b)

r (nm)

Fig. 4.3 Profiles of (a) density and (b) pressure in the nanodroplet–vapor system at 85 K and Rs = 2.99 nm

4 Vapor Pressure, Surface Tension, and Evaporation Coefficient

Density ρ (kg/m3)

120

(a)

pr

Pressure p (MPa)

pθ pφ

Eq. (4.3) (b) r (nm)

Fig. 4.4 Profiles of (a) density and (b) pressure in the nanodroplet–vapor system at 85 K and Rs = 1.43 nm

4.4.3 Pressure Distributions The distributions of pressures pr , pθ , and pφ for the nanodroplets (or px , p y , and pz for the liquid films) are shown in Figs. 4.3, 4.4, 4.5, 4.6, 4.7, and 4.8. In the following, pr (or pz ) is referred to as the normal pressure, pθ and pφ (or px and p y ) are referred to as the tangential pressures. It should be noted that the pressures pθ and pφ (or px and p y ) are not the so-called tangential stresses in fluid dynamics. As shown in Figs. 4.3, 4.4, 4.5, 4.6, 4.7, and 4.8, the tangential pressures are identical with the normal pressure in the bulk vapor and the bulk liquid, sufficiently outside the transition layer. Therefore, we regard the three pressures in the bulk liquid as the liquid pressure p L and the three pressures in the bulk vapor as the vapor pressure p V , respectively. The liquid pressure p L in Table 4.1 represents the pressure at r = 0, calculated with the use of the cubic spline interpolation with zero gradient at r = 0. The vapor pressure pV in Table 4.1 is the space-averaged pressure in the bulk vapor.

Equilibrium States of Nanodroplets and Planar Liquid Films

Density ρ (kg/m3)

4.4

121

Pressure p (MPa)

(a)

pr pθ pφ

Eq. (4.3) (b) r (nm)

(a)

pr

Pressure p (MPa)

Density ρ

(kg/m3)

Fig. 4.5 Profiles of (a) density and (b) pressure in the nanodroplet–vapor system at 105 K and Rs = 2.99 nm

pθ pφ

Eq. (4.3)

(b)

r (nm)

Fig. 4.6 Profiles of (a) density and (b) pressure in the nanodroplet–vapor system at 105 K and Rs = 1.53 nm

4 Vapor Pressure, Surface Tension, and Evaporation Coefficient

(a)

Pressure p (MPa)

Density ρ (kg/m3)

122

px py pz (b) z (nm)

(a)

Pressure p (MPa)

Density ρ (kg/m3)

Fig. 4.7 Profiles of (a) density and (b) pressure in the liquid film–vapor system at 85 K

px py pz (b) z (nm)

Fig. 4.8 Profiles of (a) density and (b) pressure in the liquid film–vapor system at 105 K

4.4

Equilibrium States of Nanodroplets and Planar Liquid Films

123

As shown in the figures, the liquid pressure p L is larger than the vapor pressure pV by the effect of surface tension for each nanodroplet, while p L is equal to pV for the liquid films. Inside the transition layer, the tangential pressures are negative and have minima in cases of both the nanodroplets and liquid films. Table 4.1 shows that the vapor pressure p V surrounding each nanodroplet increases with decreasing radius at the specified temperature, and this tendency qualitatively agrees with the results of preceding studies [8, 13, 16].

4.4.4 Differentiability of Normal Pressure with Respect to Radial Coordinate Both tangential pressures pθ and pφ are expressed to be pT in the following. The direct computation of pT was not successfully accomplished in the preceding studies [8, 13] because of the shortage of the number of sample data. Instead, pT has been obtained as follows. First, the equation of force balance, i.e., pT = pr +

r d pr , 2 dr

(4.3)

was assumed to hold in an infinitesimal volume element taken naturally in the spherical polar coordinates, as shown in Fig. 4.9. The volume element is the truncated corn with the thickness of Δr in Fig. 4.9 a, b. Equation (4.3) can be derived as follows. As shown in Fig. 4.9b, the pressure pr acts upward perpendicularly on the lower surface at r , pr (r + Δr ) acts downward on the upper surface at r + Δr , pT acts inward on the side from r to r + Δr . The balance equation of the forces in the r -direction due to pr (r ), pr (r + Δr ), and pT is given by

pr(r+Δr)

r

Δr

pT

Δr Δθ

r

pr(r)

Fig. 4.9 The balance of forces acting on the control volume; (a) a small corn in a sphere and (b) its enlargement

124

4 Vapor Pressure, Surface Tension, and Evaporation Coefficient

pr (r ) · πr 2 sin2Δθ − pr (r + Δr ) · π(r + Δr )2 sin2Δθ + pT · π [(r + Δr )2 − r 2 ] sin2Δθ = 0.

(4.4)

Putting Δr → 0 in Eq. (4.4), we obtain d 2 (r pr ) = 2r pT , dr

(4.5)

i.e., Eq. (4.3). It should be emphasized that it has not so far been confirmed whether Eq. (4.3) holds in nanoscale. It is almost self-evident that macroscopic physical quantities defined by long time average are the continuous functions of r ; however, it is not necessarily self-evident that they are differentiable with respect to r . The solid curves drawn in Figs. 4.3, 4.4, 4.5, and 4.6 are the fitted curves by use of a cubic spline interpolation of pT -data calculated with Eq. (4.3). It is evident that the tangential pressures pθ and pφ , obtained directly by the MD simulations, are in good agreement with those obtained indirectly by Eq. (4.3); this fact shows that Eq. (4.3) holds in nanoscale. The present conclusion is not affected by the thickness of the control volume. It is needless to say that Eq. (4.3) is not applicable to the liquid films whose curvature radius is infinity.

4.4.5 Laplace Equation and Surface Tension Let us obtain the surface tension γ and the position Rs (or Z s ) of the surface of tension from the pressure distributions. The surface of tension is defined as the surface on which the surface tension acts. The nanodroplet radius is defined as Rs , the position of the surface of tension. In the stationary system of a nanodroplet and its surrounding vapor in an equilibrium state, γ and Rs are determined as follows. Figure 4.10 shows the control surface ABCD to determine γ and Rs . The pressures outside the arc AB with the radius RV are the uniform vapor pressure pV . The pressures inside the arc DC with the radius R L are the uniform liquid pressure p L . Let us define an arc with radius Rs , within and outside which the pressures are p L and pV , respectively. The surface tension acts on this arc in the direction shown in the figure. The resultant force of the forces due to the liquid pressure p L , the vapor pressure pV , and the surface tension γ can be put to be equivalent with the force due to the tangential pressure pT , and therefore the balance of the forces is given by [9]

Rs 2π

p L ·r dθ dr − γ ·2π Rs +

RL 0

from which we obtain

RV 2π Rs

0

pV ·r dθ dr =

R V 2π RL 0

pT ·r dθ dr, (4.6)

4.4

Equilibrium States of Nanodroplets and Planar Liquid Films

125 r

pV

γ

RV Rs dr

RL dθ

pL

r

Fig. 4.10 The control surface ABCD for determination of γ and Rs

Rs

0

RV

p L r dr +

RV

pV r dr − γ Rs =

pT r dr,

Rs

(4.7)

0

where the position RV is arbitrarily chosen in the bulk vapor sufficiently away from the transition layer, as long as the pressures at r = RV and in the region 0 r R L are isotropic. Here, the averaged tangential pressure ( pθ + pφ )/2 is used for pT in Eq. (4.7), although either pθ or pφ can be used. Simpson’s formula is used for integration. The equation for balance of moments can also be obtained to be [9]

Rs

p L r dr + 2

0

RV

p V r dr − γ 2

Rs

Rs2

=

RV

pT r 2 dr.

(4.8)

0

Similarly, the surface tension γ∞ and the position Z s of the surface of tension in the case of a liquid film are determined by the following two equations, i.e., the equations for the balance of forces and for the balance of moments:

Zs

p L dz +

0

0

Zs

ZV Zs

p L zdz +

ZV Zs

pV dz − γ∞ =

ZV

pT dz,

pV zdz − γ∞ Z s =

(4.9)

0

ZV

pT zdz,

(4.10)

0

where the definition of Z V is the same as that of R V as mentioned above, and ( pθ + pφ )/2 is used for pT in Eqs. (4.9) and (4.10). First, eliminating γ from Eqs. (4.7) and (4.8), we obtain a cubic equation on Rs , where it should be noted that p L and p V are constant. The solution of this

126

4 Vapor Pressure, Surface Tension, and Evaporation Coefficient

cubic equation to be sought is a real and positive value among three solutions and is defined as Rs . Consequently, γ is obtained by substituting Rs back into either Eq. (4.7) or Eq. (4.8). There also exists the alternative to determine γ and Rs by the aid of Eq. (4.3) whose validity has been already confirmed. Eliminating pT from the right-hand side of Eq. (4.7) by using Eq. (4.3), we obtain the well-known Laplace equation: p L − pV =

2γ . Rs

(4.11)

No assumption is introduced in order to derive Eq. (4.11) because Eq. (4.7) used for the derivation of Eq. (4.11) is the universal law of the force balance. Although the presupposition made in the above discussion is the equality of the normal and tangential pressures at the center of the nanodroplet, this presupposition is naturally expected. Therefore, the proof of the validity of Eq. (4.3) provided in Sect. 4.4.4 is essentially equivalent with the proof of Eq. (4.11). It should be emphasized that the validity of Eq. (4.11) in nanoscale has been successfully proven on the basis of the force balance, although Eq. (4.11) can be derived from thermodynamics [9]. Secondly, eliminating pT in the right-hand side of Eq. (4.8) by using Eq. (4.3), we obtain Rs

3

3 = p L − pV

RV

r 2 ( pr − pV )dr.

(4.12)

0

Substituting Rs obtained from Eq. (4.12) into Eq. (4.11), we can determine γ . The cubic equation on Rs explained above reduces to Eq. (4.12). The values of γ and Rs evaluated are shown in Table 4.1. It is clearly understood that the surface tension γ decreases with decreasing Rs . On the other hand, γ∞ and Z s , in the case of liquid films, are evaluated from Eqs. (4.9) and (4.10), and values of them are shown in Table 4.2. The values of surface tension decrease with increasing temperature and quantitatively agree well with the experimental values shown in the parentheses [6, 12].

4.4.6 Kelvin Equation The thermodynamic condition for a single-component vapor–liquid two-phase system in an equilibrium sate is described by − sV dT + vV d pV = −s L dT + v L d p L ,

(4.13)

where s is the entropy per unit mass, T is the temperature of the system, v is the specific volume, and the subscripts V and L denote the vapor and liquid, respectively. Eliminating d p L from Eq. (4.13) with the use of differentiation of Eq. (4.11), we obtain

4.4

Equilibrium States of Nanodroplets and Planar Liquid Films

d pV =

127

2γ sV − s L vL . dT + d vV − v L vV − v L Rs

(4.14)

The first term of the right-hand side in Eq. (4.14) means the contribution of temperature change to the vapor pressure and the second term means the contribution of changes in both the radius and surface tension of the nanodroplet, d(2γ /Rs ). In the liquid film for Rs → ∞, Eq. (4.14) reduces to the well-known Clausius–Clapeyron equation. Approximating vV − v L by vV because of vV v L in Eq. (4.14), disregarding a change in v L , and integrating under a constant temperature, i.e., dT = 0, we obtain

2γ pV = p∞ exp ρ L RT Rs

,

(4.15)

where we have used the state equation, pV vV = RT , and v L = 1/ρ L , and determined an integrating constant with the use of the saturated vapor pressure p∞ at Rs → ∞. Equation (4.15) is called the Kelvin equation [9]. The comparison between the vapor pressure p V obtained by the MD simulations and that obtained by the Kelvin equation (4.15) are shown in Fig. 4.11. The values of surface tension γ , saturated vapor pressure p∞ in the case of liquid film, and liquid density ρ L , obtained by the MD simulations are used in Eq. (4.15). Figure 4.11 shows that the Kelvin equation (4.15) agrees well with the results of MD simulations except for the cases either of nanodroplets smaller than about 3.0 nm in radius at 105 K or about 1.5 nm at the other temperatures. It should be noted that the Kelvin equation is the thermodynamic equation for the vapor–liquid system in the equilibrium state, and the validity of this equation applied to nanodroplets is con-

Vapor Pressure pV (MPa)

Eq. (4.15)

Droplet Radius Rs (nm)

Fig. 4.11 The vapor pressure as the function of nanodroplet radius

105 K 100 K 95 K 90 K 85 K

128

4 Vapor Pressure, Surface Tension, and Evaporation Coefficient

firmed quantitatively, as long as nanodroplets are larger than a certain size. However, quantitative agreement between the MD simulations and the Kelvin equation is not observed for nanodroplets with Rs 3 nm at 105 K and also for extremely small nanodroplets with Rs 1.5 nm. In the following, we will discuss the reason for the disagreement and the limitation in applying thermodynamics to nanodroplets by using the results at 85 K. Equation (4.13) is obtained from the equation of balance of chemical potentials for the bulk vapor and bulk liquid, and also is the fundamental equation of thermodynamics in phase equilibrium system. Assuming the quasistatic change in radius of a nanodroplet under a constant temperature, i.e., dT = 0, we can rewrite Eq. (4.13) as ρL d pL = , d pV ρV

(4.16)

where ρ L = 1/v L and ρV = 1/vV . Let us discuss applicability of Eq. (4.16) to the nanodroplets. As shown in Fig. 4.12a, a cubic function is fitted on the data of p L , and drawn as the solid line against pV . The derivative of this cubic function is d p L /d p V , and drawn as the solid line along with the data of ρ L /ρV , which are shown as open circles, against pV in Fig. 4.12b. The fit of ρ L /ρV data to d p L /d pV curve is good only in the region of pV smaller than 0.15 MPa. This result implies that Eq. (4.16) does not hold for pV 0.15 MPa. On the other hand, an approximation function on the data of ρ L /ρV against pV is obtained by means of a nonlinear regression, and drawn as the dashed line in Fig. 4.12b. The integration of ρ L /ρV is calculated from this approximation function and drawn as the dashed line in Fig. 4.12a; an integrating constant is determined so that the dashed line may pass the point of pV = p L = 0.083 MPa. This point corresponds to the condition for liquid film (Rs → ∞). The dashed line shown in Fig. 4.12a also deviates from the data of p L for pV 0.15 MPa, i.e., the dashed line monotonously increases, while the data of p L show a convex with the ρL

∫ρ

V

dpV + C

85 K

pL/pV

pL (MPa)

85 K

dρL dρV

(a)

(b) pV (MPa)

Fig. 4.12 The verification of Eq. 4.16

pV (MPa)

4.4

Equilibrium States of Nanodroplets and Planar Liquid Films

129

maximum; hence, they disagree qualitatively. It should be noted that the larger is pV , the smaller is the droplet radius Rs ; pV 0.15 MPa corresponds to Rs 0.15 nm. From the above discussion, we can conclude that Eq. (4.16), i.e., thermodynamic equilibrium condition (4.13) does not hold for Rs 1.5 nm at 85 K, that is, not only the Kelvin equation (4.15) but also any thermodynamics theories based on Eq. (4.13) do not hold in this region. Although a theoretical explanation for the above results is not yet available, it is suggested that the discussion based on chemical potentials for the bulk vapor and bulk liquid is invalid, when the number of molecules in the liquid phase is not so different from that in the transition layer. We may conceive that the existence of the transition layer should be taken into account in the theoretical explanation. As shown in Figs. 4.5 and 4.6, the thickness of the transition layer and the number of molecules contained there at 105 K are larger than those at 85 K. The deviation of the Kelvin equation (4.15) from the MD results at 105 K is larger than that at 85 K for nanodroplets of the same radius, for example, Rs ∼ = 2.3 nm in Fig. 4.11.

4.4.7 Tolman Equation Finally, let us consider once again the vapor–liquid system of a nanodroplet and its surrounding vapor in an equilibrium state. Then, the relation between the surface tension γ and the droplet radius Rs is given by the Tolman [15]: 1 2 1 + (Δ/Rs ) + (Δ/Rs ) 1 dγ 3 , = 1 γ dRs 2 1 + (2Δ/Rs ) 1 + (Δ/Rs ) + (Δ/Rs ) 3 (2Δ/Rs2 )

(4.17)

where Δ is the physical quantity called the Tolman length and it is defined as Δ = R 0 − Rs ,

(4.18)

where R0 is the position of the Gibbs dividing surface (equimolor dividing surface) and Rs is that of the surface of tension on which the surface tension acts. For the droplet–vapor system consisting of N molecules in the cubic of L 3 in volume, R0 is given by [8] 4 4 3 3 3 m N = π R 0 ρ L + L − π R0 ρ V . 3 3

(4.19)

For the liquid film–vapor system, the position Z 0 of the dividing surface is given by m N = 2ρ L Z 0 L x L y + ρV (L z − 2Z 0 )L x L y .

(4.20)

4 Vapor Pressure, Surface Tension, and Evaporation Coefficient Surface Tension γ (N/m)

130

Eq. (4.17)

85 K 105 K

Droplet Radius Rs (nm)

Fig. 4.13 The surface tension as the function of nanodroplet radius

Disregarding Δ/R0 and (Δ/R0 )2 compared with unity in Eq. (4.17) and integrating it, we obtain γ =

γ∞ , 1 + (2Δ/Rs )

(4.21)

where an integrating constant is determined from γ = γ∞ at Rs → ∞. Although this equation is usually called the Tolman equation, it is clear that the Eq. (4.17) is more fundamental than Eq. (4.21). Thus, the validity of Eq. (4.17) for nanodroplets has been investigated in this subsection. Figure 4.13 shows the comparison between the results of MD simulations and the integration of Eq. (4.17) for the surface tension γ at 85 and 105 K. The RungeKutta method is used for the integration of Eq. (4.17); the sufficiently large radius Rs (=1 m) and γ∞ , the surface tension of liquid film, are given as initial conditions. Equation (4.17) is integrated with the dependence of the Tolman length Δ on Rs , defined by Eq. (4.18). From Fig. 4.13, it is found that the surface tension obtained by MD simulations quantitatively disagrees with the one evaluated by Eq. (4.17). Although Eq. (4.13) is quantitatively correct in the region of Rs ∼ = 4 nm as discussed in the preceding subsection, Eq. (4.17) does not hold in this region. Further investigation on reason of this result should be a target of the future work.

4.5 Mass Transport Across Nanodroplet Surface 4.5.1 Problem Statement In the previous sections, the vapor pressure and surface tension of argon nanodroplets have been carefully dealt with for extremely small radii ranging from 1 to 4 nm at the equilibrium states of temperature 85, 90, 95, 100, and 105 K by the MD simulations, and the droplet radii for which thermodynamics fails, have been clarified. However, they are static properties in the equilibrium states. In this section, we study dynamical properties such as mass transport between a nanodroplet

4.5

Mass Transport Across Nanodroplet Surface

131

and its surrounding vapor in nonequilibrium states by using nonequilibrium MD simulations. We shall clarify how the evaporation coefficient depends on the droplet radius. However, the evaluation of the evaporation coefficient depends on not only the droplet radius but also the position of control surface where the evaporation coefficient is evaluated. Thus, we shall discuss how the control surface should be chosen suitably for correct evaluation of the evaporation coefficient. The specific surfaces such as the Gibbs dividing surface and the surface of tension are known to be defined in the vapor–liquid transition layer [9, 13]. In a similar way, we shall define a new specific surface where the mass flux of molecules spontaneously evaporating from the nanodroplet is equal to that from the planar liquid film, and show that such a surface is located at a position of about one nanometer from the center of the transition layer although the position depends on the liquid temperature. Using this surface, we successfully determine the evaporation coefficient, finally obtaining a formula which describes the dependence of the evaporation coefficient on the droplet radius and the liquid temperature.

4.5.2 Evaporation and Condensation Coefficients, and Mass Transfer Rate Definitions of mass fluxes Jevap , Jcoll , Jcnds , Jref , and Jcnds (= Jcoll − Jref ) are the same ones as in Sect. 3.1 and they also hold for nanodroplets. The mass fluxes Jout and Jcoll are evaluated at the control surface with radius r in the spherical coordinate system, whose origin coincides with the gravity center of the nanodroplet. We can classify molecules passing through the control surface into two groups; (i) molecules going out from the control surface into the vapor and (ii) molecules from the vapor into the control surface. Let ΔNout be the number of the former molecules counted during a short time Δt in the MD simulations as shown in the previous sections, and ΔNcoll be the latter molecules during Δt. The fluxes Jout and Jcoll are then obtained by Jout =

mΔNout , SΔt

Jcoll =

mΔNcoll , SΔt

(4.22)

where m is the mass of one molecule, S is the area of the control surface, and the angle brackets denote the sample average. The flux Jevap is a part of the flux Jout and it is defined as the flux depending on the liquid temperature only and it is defined as the mass flux of spontaneously evaporating molecules from the liquid. The flux Jevap is thus independent of the flux Jcoll . The flux Jevap is obtained by nonequilibrum MD simulations of the virtual vacuum evaporation, in which molecules going out from the control surface into the vapor never return to it, as will be explained in the next subsection. The flux Jref is defined by Jref = Jout − Jevap . Furthermore, the flux Jref leads to the proper

132

4 Vapor Pressure, Surface Tension, and Evaporation Coefficient

definition of the flux Jcnds by Jcnds = Jcoll − Jref . The mass conservation law at the interface leads to Jout = Jevap + Jref ,

Jcoll = Jref + Jcnds .

(4.23)

The evaporation coefficient αe and the condensation coefficient αc are defined by Eq. (3.1). Making use of these definitions, the net mass transfer rate per unit time between a nanodroplet and its surrounding vapor is given by RTL dM 2 = 4πr αc Jcoll − αe ρV , dt 2π

(4.24)

where M is the mass of the nanodroplet, t is√the time, and r is the radius of the control surface. If we replace Jcoll by ρv∞ RTv∞ /(2π ), where ρv∞ and Tv∞ are respectively the vapor temperature and density sufficiently far from the control surface, and αc and αe by α, we have the well-known Hertz–Knudsen–Langmuir formula [1]. However, this formula is derived under the assumption that the velocity distribution function of molecules colliding against the interface from the vapor is the Maxwellian and that αc equals to αe even in nonequilibrium states, and it is incorrect, as criticized in Sect. 2.6.

4.5.3 Vacuum Evaporation Simulations The mass flux Jevap is identical with the flux Jout for the virtual vacuum evaporation because Jref = 0 in this case. Nonequilibrium MD simulations of the virtual vacuum evaporation of single argon nanodroplets with 1–4 nm in radius are performed to evaluate the mass flux Jevap at 85, 90, 95, 100, and 105 K. The simulations are preformed by extending for the planar interface [2, 3] to the case of the spherical interface. Intermolecular potential is the Lennard-Jones 12-6 potential with the same parameters as in the equilibrium simulations. Table 4.3 shows the initial radius Rm0 and the initial 10–90 thickness δ0 of nanodroplets for the virtual vacuum evaporation at 85, 90, 95, 100, and 105 K, where Rm0 is an equilibrium value of the distance from the center of the droplet to the center of the transition layer, as shown in Fig. 4.14. The results of the equilibrium Table 4.3 The initial radius Rm0 and initial 10–90 thickness δ0 of the transition layer of argon nanodroplet in simulations of the virtual vacuum evaporation at TL TL (K) Rm0 (nm) δ0 (nm) 85 90 95 100 105

4.31 4.23 4.10 3.90 3.58

0.72 0.79 0.86 0.91 0.98

4.5

Mass Transport Across Nanodroplet Surface

133

Liquid

δ

Temperature Control Region

ρL

Boundary Condition of Vacuum

ρ

Vapor

O R c Rm (Zc) (Zm)

r (z)

Rg (Zg)

Fig. 4.14 The boundary condition of vacuum for the virtual vacuum evaporation of argon nanodroplet and the temperature control region. In the figure, ρ L and ρV in the equilibrium state are shown

simulations are used as the initial conditions. We carry out 8 simulations at each temperature and physical quantities to be obtained are averaged to make statistical errors as small as possible. The initial configurations for the 8 cases are different in microscopic quantities, i.e., molecular positions and velocities, but macroscopic quantities such as Rm0 , δ0 , and TL are almost the same. We use the data obtained after 0.12 ns from the start of simulations, thereby eliminating the influence of the initial conditions. It is also confirmed that results such as Jevap are not affected by different initial conditions such as Rm0 . Figure 4.14 shows the spherical boundary condition for virtual vacuum with the radius Rg and the temperature control region with the radius Rc in the simulations. The virtual vacuum is set in the region of r Rg and the evaporation into the virtual vacuum is realized by deleting of molecules going out across the boundary. The radius Rg is determined to be finally Rg = Rm + 5δ0 by trial and error; other cases, e.g., Rm + 3δ0 and Rm + 7δ0 , give the same results in Jevap as Rg = Rm + 5δ0 . The latent heat is robbed from the liquid due to the evaporation. Therefore, the velocity scaling is applied to all molecules in the region of 0 r Rc at every time step to keep the liquid temperature TL constant; Rc is defined to be Rm −0.5δ0 as shown in Fig. 4.14. It should be noted that the radius Rm decreases with the lapse of time as shown in Fig. 4.15, and in consequence, Rg and Rc also decrease. Density distributions in the radial direction are obtained every 0.12 ns and Rm , Rg , and Rc are recalculated. For liquid films, Z g and Z c are used in place of Rg and Rc , as shown in Fig. 4.14.

4.5.4 Mass Fluxes and Evaporation Coefficient √ Figure 4.16 shows the dependence of the mass flux ρV RTL /(2π ) of molecules flying out from droplet of radius Rs ranging from about 1 to 4 nm in the equilibrium

4 Vapor Pressure, Surface Tension, and Evaporation Coefficient

85 K 90 K 95 K 100 K 105 K

Rm (nm)

134

Time (ns)

Fig. 4.15 Changes in time of radius Rm in the virtual vacuum evaporation. The temperature shown in the figure denotes the liquid temperature TL kept constant by the velocity scaling method

states at 85, 90, 95, 100, and 105 K; the radius Rs is defined to be that of the surface of tension √ [16] and the relation between Rm and Rs is shown in Fig. 4.17. The mass flux ρV RTL /(2π ) increases with decreasing Rs for all cases of TL as shown in Fig. 4.16, because the vapor density ρ V exponentially increases with decreasing Rs according to the Kelvin equation (4.15). The flux is also dependent on the temperature TL ; at a fixed radius, the flux increases with increasing TL . Figure 4.18a, b show the profiles of the mass fluxes Jout and Jcoll in the vacuum evaporation states at TL = 85 and 105 K. The spherical control surface across which the fluxes are evaluated is set so that its center can coincide with the gravity center of the nanodroplet each. In the vapor region, the flux Jcoll is much smaller than

V

RTL /(2 ) [kg/(m2s)]

85 K 90 K 95 K 100 K 105 K

Droplet Radius Rs (nm)

Fig. 4.16 Relations between ρV 85, 90, 95, 100, and 105 K

√

RTL /(2π ) and Rs for nanodroplets in the equilibrium states at

4.5

Mass Transport Across Nanodroplet Surface

135

85 K

Rs (nm)

105 K

Rm (nm)

Fig. 4.17 The relation between Rs and Rm in the equilibrium states at 85 and 105 K

the flux Jout , and therefore, the virtual vacuum evaporation condition Jcoll = 0 is satisfied; the flux Jout in the vapor region is identical with the flux Jevap . The flux Jevap slightly decreases with increasing r since it is a mass flow per unit area. In the following, we shall consider the most reasonable choice of the radius of control surface to evaluate the flux Jevap . Figure 4.19a, b show the dependence of the outgoing mass flow rate Q out on the spherical control surface at the position r under the same conditions as in Fig. 4.18a, b, where Q out is defined by Q out = 4πr 2 Jout .

(4.25)

Jout(Jevap) Jcoll

85 K

(a)

Rm = 3.40 nm

r (nm)

Mass Flux [kg/(m2s)]

Mass Flux [kg/(m2s)]

We can see that Q out is constant, i.e., Jout = Jevap , for r 4.5 nm at 85 K and r 3.5 nm at 105 K. We can therefore define an evaporation mass flow rate Q evap

Jout(Jevap) Jcoll

105 K

(b)

Rm = 2.64 nm

r (nm)

Fig. 4.18 Profiles of the outgoing mass flux Jout and the incoming flux Jcoll in the virtual vacuum evaporation of nanodroplets at (a) 85 K and (b) 105 K

85 K

Qout [(kg/s) 10

Qout [(kg/s) 10

21]

4 Vapor Pressure, Surface Tension, and Evaporation Coefficient

21]

136

Qevap

(a)

Rm = 3.40 nm

r (nm)

105 K Qevap

(b)

Rm = 2.64 nm

r (nm)

Fig. 4.19 Profiles of the outgoing mass flow rate Q out in the virtual vacuum evaporation at (a) 85 K and (b) 105 K. The solid lines represent the evaporation mass flow rate Q evap in the vapor region

by an average of Q out in a region at some distance from the droplet,where Q out is almost constant. Figure 4.20a–e show the profiles of the flux Jevap for several cases of Rm at 85, 90, 95, 100, and 105 K. The abscissas are r − Rm , the distance between Rm and r for the nanodroplets, and z − Z m , the distance between Z m and z for the planar liquid films (Rm → ∞). The dashed lines represent the flux Jevap for the ∞ ; J ∞ and other physical quantities are summarized in planar liquid films, i.e., Jevap evap Table 4.4. In the vapor-side transition layer, for example, at r − Rm ∼ = 1 nm at 85 K, the flux Jevap becomes larger as Rm becomes smaller. This is similar at the other temperatures. In the vapor region outside the transition layer, the flux Jevap rapidly ∞ -line for all temperatures decreases with increasing r − Rm , and crosses the Jevap and radii. The most noteworthy point is that the profiles of the flux Jevap for the different radii at a specified temperature intersect with each other at a certain point ∞ -line. This fact suggests that there exists a specific control surface at on the Jevap ∞ holds for a nanodroplet considered. Let R which Jevap = Jevap evap be the radius of this control surface. This radius Revap is expected as the radius of the control surface suitable for evaluating the mass flux without any ambiguity. Substituting ∞ into Eq. (4.25) gives Q out = Q evap , r = Revap , and Jout = Jevap Revap =

Q evap . ∞ 4π Jevap

(4.26)

Let ρV∞ and αe∞ be the saturated vapor density and evaporation coefficient for the planar liquid films (Table 4.4), respectively. On the basis of the above finding on coefficient αe for the nanodroplets by αe∞ . Revap , we will express the evaporation √ √ ∞ /[ρ ∞ RT /(2π )] and α = J /[ρ RT /(2π )], and Noting that αe∞ = Jevap L e evap V L V ∞ = J equating Jevap at R for the nanodroplets, we obtain evap evap αe = αe∞

ρV∞ . ρV

(4.27)

Mass Transport Across Nanodroplet Surface

∞ Jevap

Planar Interface

(a) r

90 K

(b)

Planar Interface

r

3.8 2.8 1.9

95 K

∞ Jevap

(c) r

Rm(nm)

100 K

R m (nm) 3.6 2.6 1.8

∞ Jevap

(d)

Planar Interface

4.0 3.0 1.9

∞ Jevap

Rm(nm) R m (nm)

Jevap [kg/(m2s)]

R m (nm)

Jevap [kg/(m2s)]

R m (nm) 4.0 3.0 2.0

85 K

137

Jevap [kg/(m2s)]

Jevap [kg/(m2s)]

4.5

Rm(nm)

Planar Interface

r

Rm(nm)

Jevap [kg/(m2s)]

R m (nm)

105 K

3.3 2.6 2.1

∞ Jevap

(e)

Planar Interface

r

Rm(nm)

Fig. 4.20 Profiles of Jevap versus r − Rm at 85, 90, 95, 100, and 105 K. The dashed lines represent ∞ (J Jevap evap for the planar liquid films)

TL

Table 4.4 Physical quantities for the planar liquid films at 85, 90, 95, 100, and 105 K ) ∞ L pV∞ ρV∞ ρ L∞ γ∞ ρV∞ RT Jevap 2π

(K)

(MPa)

(kg/m3 )

85 90 95 100 105

0.083 0.141 0.225 0.338 0.484

4.76 7.81 12.03 17.40 24.47

(kg/m3 )

(N/m)

[kg/(m2 s)]

[kg/(m2 s)]

αe∞

1394 1362 1329 1297 1263

0.0136 0.0124 0.0111 0.0100 0.0088

255 427 676 1003 1440

229 380 587 845 1155

0.90 0.89 0.87 0.84 0.80

138

4 Vapor Pressure, Surface Tension, and Evaporation Coefficient

We can understand that αe is a function of TL and Rs , because αe∞ and ρV∞ are dependent on TL only, and ρV is dependent on both TL and Rs [Eq. (4.15)]. In the equilibrium state of a nanodroplet at a constant temperature, i.e., dT = 0, we have Eq. (4.16), i.e., d pL d pV = , ρV ρL

(4.28)

where pV and ρV are the pressure and density of the vapor, and p L and ρ L are those of the liquid. It should be noted that Eq. (4.28) fails for Rs 1.5 nm, as clarified in Sect. 4.4.6. Eliminating d p L from Eq. (4.28) by using the differential of the Laplace equation (4.11), d p L = d pV + d(2γ /Rs ), we obtain

1 1 − ρV ρL

d pV =

2γ 1 . d ρL Rs

(4.29)

Approximating that 1/ρV − 1/ρ L ∼ = 1/ρV and ρ L ∼ = ρ L∞ , we obtain

2γ d pV = ∞ + const. ρV ρ L Rs

(4.30)

In order to perform the integration of the left-hand side in the above equation, we shall use the Berthelot equation as the state equation of real gases [5, 10]: % ρ V ( p V , TL ) =

&−1 6Tc2 1 9Tc − B(TL ) RTL , B(TL ) = −1 , pV 128 pc TL TL2

(4.31)

where T is the gas temperature, Tc is the critical temperature, and pc is the critical pressure; Tc = 150.7 K and pc = 4.86 MPa for argon [12]. Substituting Eq. (4.31) into Eq. (4.30), and determining a constant by the condition pV → pV∞ as Rs → ∞, we obtain pV (Rs , TL ) 2γ ln − [ pV (Rs , TL ) − pV∞ ]B(TL ) = ∞ . (4.32) pV∞ ρ L RTL Rs Equation (4.32) is the thermodynamic equation which describes the dependence of pV on Rs . Let [ pV (Rs , TL ) − pV∞ ]B(TL ) in the left-hand side of Eq. (4.32) be X . Values of X are, for example, 0.028 for Rs = 1.7 nm at TL = 85 K and 0.036 for Rs = 2.3 nm at TL = 105 K, and X decreases with increasing Rs and it approaches 0 as Rs → ∞ because then pV → pV∞ . In consequence, we have the relation X ≈ − ln(1 − X ) in the range of Rs from about one nanometer to infinity at temperatures near the triple point. Substituting this relation into Eq. (4.32) yields a quadratic equation with respect to pV , and its solution is

4.5

Mass Transport Across Nanodroplet Surface

139

pV (Rs , TL ) = pV∞ F(Rs , TL ),

(4.33)

where 1 1 F(Rs , TL ) = 1+ ∞ 2 pV B(TL ) 2 1 1 4 2γ − 1+ ∞ . − ∞ exp − ∞ 2 pV B(TL ) pV B(TL ) ρ L RTL Rs In the above, the negative sign of the second term in the right-hand side is adopted so as that pV approaches pV∞ for Rs → ∞. Substituting Eq. (4.33) into Eq. (4.31), we obtain ρ V (Rs , TL ) =

ρV∞ /

%

& ρV∞ RTL 1 1− 1− . pV∞ F(Rs , TL )

(4.34)

From Eqs. (4.27) and (4.34), we finally obtain αe (Rs , TL ) =

αe∞

% & ρV∞ RTL 1 1− 1− . pV∞ F(Rs , TL )

(4.35)

The surface tension γ is included in F(Rs , T ), and hence the value of γ is needed to calculate αe (Rs , TL ). In Eq. (4.35), the real gas effect on ρV in equilibrium states is considered. If we use the state equation of ideal gases instead of Eq. (4.31), and substitute it and the Kelvin equation (4.15) into Eq. (4.27), we have αe (Rs , TL ) =

αe∞ exp

−

2γ ρ L∞ RTL Rs

.

(4.36)

It should be noted that the assumption of ideal gases, i.e., B(TL ) = 0, reduces Eq. (4.31) to the state equation of ideal gases and Eq. (4.32) to the Kelvin equation (4.15). The dashed curves in Fig. 4.21 represent αe evaluated by Eq. (4.36). At 85 and 90 K, the dashed curves are in good agreement with αe by the MD values and Eq. (4.35). The higher TL becomes, the worse the agreement is because of the real gas effect caused by the increase in pV . The disagreement between the solid curves by Eq. (4.35) and the dashed curves by Eq. (4.36) at 95, 100, and 105 K becomes smaller as Rs increases, because pV decreases. Therefore, Eq. (4.36) can be used instead of Eq. (4.35) at TL near the triple point or for sufficiently large Rs . Equation (4.35) is of importance for transport phenomena between droplets and their surrounding vapor, and it is also useful in the KBC for the Boltzmann equation and boundary conditions for the set of Navier–Stokes equations.

4 Vapor Pressure, Surface Tension, and Evaporation Coefficient

0.8

e

1.0 ∞ e

0.6

85 K 0.4

Eq. (4.35) 0.2

Eq. (4.36)

(a) 0 0.1

1

10

100

1000

Evaporation Coefficient

Evaporation Coefficient

e

140

1.0

0.8

0.6

90 K 0.4

Eq. (4.35) 0.2

0 0.1

e

0.6

95 K 0.4

Eq. (4.35) Eq. (4.36)

0

0.1

1

10

100

1000

Evaporation Coefficient

e

Droplet Radius Rs(nm)

Evaporation Coefficient

e

Evaporation Coefficient

∞ e

(c)

1

10

100

1000

Droplet Radius Rs(nm)

1.0

0.2

Eq. (4.36)

(b)

Droplet Radius Rs(nm)

0.8

∞ e

1.0

0.8

∞ e

0.6

100 K

0.4

Eq. (4.35) 0.2

Eq. (4.36)

(d) 0 0.1

1

10

100

1000

Droplet Radius Rs(nm)

1.0

0.8

∞ e 0.6

105 K 0.4

Eq. (4.35) 0.2

Eq. (4.36)

(e) 0 0.1

1

10

100

1000

Droplet Radius Rs(nm)

Fig. 4.21 The dependence of the evaporation coefficient αe on the droplet radius Rs . Solid and dashed curves are theoretical predictions by Eqs. (4.35) and (4.36), respectively. A thin solid line denotes αe∞

References 1. E.J. Davis, A history and state-of-the-art of accommodation coefficients. Atmos. Res. 82, 561–578 (2006) 2. T. Ishiyama, T. Yano, S. Fujikawa, Molecular dynamics study of kinetic boundary condition at an interface between argon vapor and its condensed phase. Phys. Fluids 16, 2899–2906 (2004)

References

141

3. T. Ishiyama, T. Yano, S. Fujikawa, Molecular dynamics study of kinetic boundary condition at an interface between polyatomic vapor and its condensed phase. Phys. Fluids 16, 4713–4726 (2004) 4. T. Ishiyama, T. Yano, S. Fujikawa, Kinetic boundary condition at a vapor-liquid interface. Phys. Rev. Lett. 95, 084504 (2005) 5. J.D. Lambert, G.A.H. Roberts, J.S. Rowlinson, V.J. Wilkinson, The second virial coefficients of organic vapours. Proc. R. Soc. Lond. A196, 113–125 (1949) 6. J.W. Miller, Jr., C.L. Yaws, Correlation constants for liquids: Surface tensions of liquids. Chem. Eng. 83, 127–129 (1976) 7. I. Napari, A. Laaksonen, The effect of potential truncation on the gas-liquid surface tension of planar interfaces and droplets. J. Chem. Phys. 114, 5796–5801 (2001) 8. M.J.P. Nijmeijer, C. Bruin, A.B. van Woerkom, A.F. Bakker, J.M.J. van Leeuwen, Molecular dynamics of the surface tension of a drop. J. Chem. Phys. 96, 565–576 (1992) 9. J.S. Rowlinson, B. Widom, Molecular Theory of Capillarity (Clarendon Press, Oxford, 1982) 10. A.F. Saturno, Daniel Berthelot’s equation of state. J. Chem. Edu. 39, 464–465 (1962) 11. Y. Sone, Kinetic Theory and Fluid Dynamics (Birkhauser, ¨ Boston, MA, 2002) 12. C. Tegeler, R. Span, W. Wagnera, A new equation of state for argon covering the fluid region for temperatures from the melting line to 700 K at pressures up to 1000 MPa. J. Phys. Chem. Ref. Data 28, 779–850 (1999) 13. S.M. Thompson, K.E. Gubbins, J.P.R.B. Walton, R.A.R. Chantry, J.S. Rowlinson, Molecular dynamics study of liquid drops. J. Chem. Phys. 81, 530–542 (1984) 14. W. Thomson, On the equilibrium of vapour at a curved surface of liquid. Phil. Mag. 42, 448–452 (1871) 15. R.C. Tolman, The effect of droplet size on surface tension. J. Chem. Phys. 17, 333–337 (1949) 16. J. Vrabec, G.K. Kedia, G. Fuchs, H. Hasse, Comprehensive study of the vapour-liquid coexistence of the truncated and shifted Lennard-Jones fluid including planar and spherical interface properties. Mol. Phys. 104, 1509–1527 (2006) 17. H. Yaguchi, T. Yano, S. Fujikawa, Molecular dynamics study of vapor-liquid equilibrium state of an argon nanodroplet and its vapor. J. Fluid Sci. Technol. 5, 180–191 (2010)

Chapter 5

Dynamics of Spherical Vapor Bubble

Abstract The principal object of this chapter is to present from both macroscopic and microscopic viewpoints a physically correct set of equations for thermo-fluid dynamic problems accompanied by evaporation or condensation at the interface between the bulk vapor phase and the bulk liquid phase of the same molecules. We incorporate the boundary conditions derived in Chapter 2 for vapor flows in the fluid-dynamic region outside the Knudsen layer on the interface into the set of equations; the evaporation and condensation coefficients determined in Chapter 3 can be use in the boundary conditions. We first derive general boundary conditions by considering the conservation equations of mass, momentum, and energy at the interface. We then derive a set of equations describing the dynamics of a spherical vapor bubble accompanied by the evaporation or condensation at the interface as an example of the application of the physically correct sets of equations to practically important problems. It should be emphasized most that the method enables us to solve in a mathematically rigorous manner the flow fields of both internal and external the bubble, and the dynamics at the bubble wall by taking temperature and density distributions inside the bubble into account under the assumption of uniform pressure.

5.1 Fluid-dynamic Definition of Interface Let the form of a vapor–liquid interface given by the set of points x at a time t be [1, 16] Ξ (x, t) = 0,

(5.1)

where Ξ is a scalar continuous function (see Fig. 5.1). This interface divides the whole space considered into two completely separated spaces. One is occupied by a single-component pure liquid, and the other by the pure vapor of the liquid. Now, we choose a point x p on the interface, and then consider an infinitesimally small surface element expressed as the set for points x q satisfying

S. Fujikawa et al., Vapor-Liquid Interfaces, Bubbles and Droplets, Heat and Mass C Springer-Verlag Berlin Heidelberg 2011 Transfer, DOI 10.1007/978-3-642-18038-5_5,

143

144

5 Dynamics of Spherical Vapor Bubble

Fig. 5.1 Definition of Ξ (x, t), and the unit normal in the outward direction n at x

Ξ (x q , t) = 0

q x − x p < ,

(5.2)

where is a small number and defines the neighborhood of x p . We define an infinitesimal surface element S 0 as the set of x q given by Eq. (5.2). Then the infinitesimal surface element at a time t + dt can be expressed as " ! S(t) = S 0 = S t; S 0 ,

! " S(t + dt) = S t + dt; S 0 ,

(5.3)

Equation (5.3) represents that the set of points on the interface S at the time t +dt can be exclusively written as a function of the set of points on the interface S 0 at the time t and elapsed time dt. This expression is referred to as a Lagrangian description of interface motion. It should be noted that S in Eq. (5.3) is not a function of space x, but a function of time t alone with a parameter S 0 that is an index of the surface element considered. In contrast, all points on the surface element S at t = t + dt satisfy Ξ (x θ , t + dt) = 0,

(5.4)

where the set of x θ is the infinitesimal surface element that has been the set of x q at the time t. This expression is referred to as an Eulerian form of interface motion. Hereafter S and Ξ are referred to as the Lagrangian and Eulerian descriptions of the interface, respectively, unless otherwise stated. In this chapter, we deal with the situation where the two fluids separated by the interface are pure, i.e. each of them consists of a single component. That is, the interface divides the whole space into two regions, the external region Re and the internal one Ri . Specifically, we discuss the situation in which the region Re is occupied by the pure liquid, and the region Ri is occupied by the pure vapor of the liquid. However, the following discussion throughout in Sects. 5.1, 5.2, and 5.3 is valid for the other situation in which Re is occupied by a pure vapor of liquid, and Ri by a pure liquid. Flow parameters in Re will be denoted by the superscript e, and the corresponding parameters in Ri by the superscript i.

5.2

Kinematics of Interface

145

5.2 Kinematics of Interface 5.2.1 Interface Velocity We introduce the basic concepts of differential geometry of the interface following the context of Prosperetti [19]. The normal component of the interface velocity vs at point x on the interface is obtained in the following way. Notice that Eq. (5.1) should be satisfied at the time t +dt on the point with a slight displacement vdt from the original position x due to movement of the interface. An equation of interface motion is obtained by Taylor series expansion [6]: ' ( Ξ (x + vs dt, t + dt) = dt ∂/∂t + v s · ∇ Ξ (x, t) = 0.

(5.5)

Denoting the unit normal in the outward direction at Ξ (x, t) = 0 by n, we have [16] n(x, t) = ∇Ξ (x, t)/ |∇Ξ (x, t)| ,

Ξ (x, t) = 0.

(5.6)

We obtain the normal velocity component of the interface from Eqs. (5.5) and (5.6): vs · n = −

1 ∂Ξ . |∇Ξ | ∂t

(5.7)

5.2.2 Interface Curvature We consider an infinitesimal hexahedron with volume V , whose center of mass is at a point P on a curved interface, four lateral faces are set parallel to the unit normal n P to the interface at the point P. The remaining two faces placed perpendicularly to n P . One of two faces belongs entirely to the region where Ξ > 0, and the other to the region where Ξ < 0. It should be noted that such a hexahedron can be constructed in view of the assumed regularity of Ξ ; hence the four lateral faces of the hexahedron intersect the interface. We refer to these four lateral faces as A, B, C, and D, and to corresponding unit normals directed outward to be N A , N B , N C , and N D , respectively, as shown in Fig. 5.2. We introduce a local Cartesian-orthogonal-coordinates frame with the origin at P, and with axes ξ parallel to N C , η parallel to N D , and ζ parallel to n P . The lengths of sides of the hexahedron in the directions of ξ , η, and ζ are set as dξ , dη, and dζ , respectively. Note that the localization theorem [8]1 and the Gauss divergence theorem together yield the following interpretation of the divergence [8]: Let Φ be a continuous scalar or vector field on an open set R in E , where E is*a three-dimensional Euclidean point space. Then given any x0 ∈ E , Φ(x0 ) = limδ→0 (1/vol(Ωδ )) Ωδ Φd V , where Ωδ

1

146

5 Dynamics of Spherical Vapor Bubble nA A

P

nP

NA

B

C

D

nC NB NC dϑI

ND ζ

dϑI dϑI η

dϑII ξ

Fig. 5.2 The hexahedron with infinitesimal volume ΔV

1 ∇ · n = lim V →0 V

∂V

n · Nd (∂ V ) ,

(5.8)

where N is the unit normal directed outward on the boundary ∂ V of the faces considered, and n is naturally defined as the extension of Eq. (5.6) to the case of Ξ (x, t) = 0: n(x, t) = ∇Ξ (x, t)/ |∇Ξ (x, t)|

for any Ξ (x, t).

(5.9)

Now, we calculate the right-hand side of Eq. (5.8). We evaluate the area integration of the dot products of the outward directed unit normal N to the cube and the unit normal n to the interface for all six faces. The variation of the inner product n · N evaluated on a face can be neglected since we are considering the limit of the infinitesimally small area of the face; hence the unit normal n is constant on the surface. In the limit, the contributions of the faces perpendicular to n P cancel. From the faces parallel to the plane (η, ζ ) we have the contribution to the flux given by lim

V →0

nC · N C − n A · N A dηdζ. dξ dηdζ

(5.10)

Referring to Fig. 5.2, we find that n C · N C = −n A · N A = sin dϑ I ∼ = ϑ I , and I I I dξ = 2dϑ R where R is the radius of curvature of the trace of the interface on the (ξ, ζ ) plane passing through n P . The limit of Eq. (5.10) thus reduces to 1/R I . In the same way, from the two faces parallel to the plane (ξ, ζ ), we have

(δ > 0) is the closed ball of radius δ centered at x0 . Therefore, if ball Ω ⊂ R, then Φ = 0.

* Ω

Φd V = 0 for every closed

5.2

Kinematics of Interface

147

lim

V →0

n D · N D − nB · N B dξ dζ. dξ dηdζ

(5.11)

∼ ϑ II , and dη = 2dϑ II R II where We find that n D · N D = −n B · N B = sin dϑ II = II R is the radius of curvature of the trace of the interface on the (η, ζ ) plane passing through n P . The limit of Eq. (5.11) thus reduces to 1/R II . From Eqs. (5.10) and (5.11), we finally obtain the following relation: ∇·n=

1 1 + II . RI R

(5.12)

It should be noted that the orientation of the (ξ, η) axes in the tangent plane to the interface at the point P used here is arbitrary. Consequently, we obtain the important feature of Eq. (5.12) that (1/R I + 1/R II ) is indifferent to rotation of the hexahedron around n P , although values of R I and R II vary in general. The total curvature C of the interface at the point P is defined by C = 1/R + 1/R ,

(5.13)

where R and R are the radii of curvature of the section of the interface with any two planes orthogonal to each other and to the interface at P. The family of surfaces Ξ = constant defines, by means of Eq. (5.6), a vector field of normals; hence with the use of Eq. (5.12), the total curvature C can be written as C =∇·n

Ξ = 0.

on

(5.14)

5.2.3 Time Variation of Area of Surface Element The interface is a function of time t, as expressed by Eqs. (5.1) or (5.3) in the Lagrangian description. For further discussion of the interface motion, the increase in per unit surface area and unit time (1/Ξ )(DΞ/Dt) should be obtained. First, we consider a volume V(t), in general, bounded by a closed surface S(t). On the surface, the unit normal N is taken outward, and the velocity field both on the surface and inside the volume V(t) is vs (x, t). Now, we consider the volume that the closed surface S sweeps out during an infinitesimal time Δt. With the use of the Gauss divergence theorem, this volume can be calculated:

Δt

S

vs · NdSdτ =

Δt

V

∇ · vs dVdτ.

(5.15)

We notice that the total volume V consists of the summation of each infinitesimally * small volume element dV i.e., V = V dV. We, here, consider infinitesimally small volume element V (t) bounded by a closed surface S(t), instead of dV. Then, Eq. (5.15) can be rewritten as

148

5 Dynamics of Spherical Vapor Bubble

1 dV (τ ) = ∇ · vs . V (τ ) dτ

(5.16)

The left-hand side of Eq. (5.16) is the increment of volume element due to the surface movement during an infinitesimally small time dt divided by V ; hence Eq. (5.16) can be rewritten as V (t + dt) = V (t) + ∇ · vs dt V (t) = (1 + ∇ · vs dt) V (t).

(5.17)

Let us consider the special case that the infinitesimal volume element V (t) is in a form of cylinder orthogonal to the interface and protruding equally on either side (interior and exterior) of the interface. Let an intersectional surface between V (t) and the interface be an infinitesimal surface element S(t). The peripheral edge of this surface element S(t) is assumed well-smooth-closed line and defined as C, and the origin of ζ -coordinate is located on the interface. It should be noticed here that this infinitesimal surface element S(t) is a function of time in the Lagrangian description as defined in Eq. (5.3). We notice that the closed surface enclosing the volume element V in the form of cylinder as shown in Fig. 5.3 can be divided into a lateral surface and a pair of upper and lower base-plane surfaces. The lateral surface of this cylinder is defined as S s . The upper surface of this cylinder in the external side is defined as S e , located on the coordinate ζ of ζ e , and similarly the lower surface is defined as S i , located on the coordinate ζ of ζ i , where the superscript e (for external side) denotes quantities evaluated on the positive side of Ξ and the superscript i (for internal side) denotes quantities evaluated on the negative side of Ξ . The unit normals to these bases are defined as N e = n |ζ =ζ e ,

and

N i = −n |ζ =ζ i ,

Ne Se | ζ e|

N S

h C

nC

| ζi | Si

ζ

η

Ni ξ

Fig. 5.3 The infinitesimal volume element

(5.18)

5.2

Kinematics of Interface

149

where n is defined by Eq. (5.9). The height of the cylinder, h, is given by h = ζe − ζi.

(5.19)

Now, the volume element V (t), as the one shown in Fig. 5.3, can be simply expressed by V (t) = h(t)S(t).

(5.20)

Substituting Eq. (5.20) into Eq. (5.17), we obtain S(t + dt) = h(t)S(t)(1 + ∇ · vs dt)/ h(t + dt).

(5.21)

We assume that the velocity vs is continuous and differentiable at the interface, then the following expansions are allowed: ∂(vs · n) e ζ , ∂ζ ∂(vs · n) i ζ , = vs · n |ζ =0 + ∂ζ

vs · n |ζ =ζ e = vs · n |ζ =0 +

(5.22)

vs · n |ζ =ζ i

(5.23)

where terms of O[(ζ e )2 ] and O[(ζ i )2 ] have been neglected and the similar approximations will be made hereafter. Then, with the use of Eqs. (5.22) and (5.23), we can write h(t + dt) as h(t + dt) = ζ e (t + dt) − ζ i (t + dt) ∂(vs · n) = h(t) + h(t)dt. ∂ζ

(5.24)

Substituting Eq. (5.24) into Eq. (5.21), we obtain ∂ S(t + dt) − S(t) = ∇ · vs − (vs · n) S. dt ∂ζ

(5.25)

Here, it should be noted that S appearing in Eq. (5.25) is in the Lagrangian description as in Eq. (5.3). We had better rewrite Eq. (5.25) in the Eulerian description for convenience in deriving further equations; hence, in the limit dt → 0, the left-hand side of Eq. (5.25) is replaced by the convective derivative of Ξ following the motion of the interface, DΞ/Dt: 1 DΞ = ∇ · vs − (n · ∇)(vs · n), Ξ Dt

(5.26)

150

5 Dynamics of Spherical Vapor Bubble

with D/Dt = ∂/∂t + vs · ∇.

(5.27)

The fact that the differentiation of a scalar with respect to ζ is equivalent to the projection of the gradient of this scalar in the direction of the normal to the interface is used. For further investigation of the right-hand side of Eq. (5.26), the surface divergence should be discussed.

5.2.4 Surface Divergence Now, we consider a vector j which is continuous and differentiable at the interface, and apply the Gauss divergence theorem to this vector field to j : j · NdS V = ∇ · j dV, (5.28) SV

V

where S V denotes the closed surface of a volume V and N is the unit normal directed outward at an arbitrary point on S V . Here, we use the same infinitesimal cylindrical volume element as that introduced in the previous section; then the surface can be divided into a lateral surface S s and a pair of upper and lower base-plane surfaces S e , S i . The unit normals to bases have already defined in Eq. (5.18), and the unit normal to the lateral surface is defined as N s . The left-hand side of Eq. (5.28) is rewritten as V e e i i j · NdS = j · N dS + j · N dS + j · N s dS s SV Se Si Ss . = j · nC dl + O(h 2 ), ( j · n)ζ =ζ e dS e − ( j · n)ζ =ζ i dS i + h Se

Si

C

(5.29) with the use of the definition of the height of the infinitesimal cylinder h (given by Eq. (5.19)), where nC is the unit normal directed outward to C, which has been already defined in the previous subsection as the peripheral edge of the intersectional surface between V (t) and the interface. Notice that since j is continuous and differentiable at the interface, the following expansions are allowed: ∂( j · n) ζ e, (5.30) ( j · n)ζ =ζ e = ( j · n)ζ =0 + ∂ζ ζ =0 ∂( j · n) ζi. (5.31) ( j · n)ζ =ζ i = ( j · n)ζ =0 + ∂ζ ζ =0

We first evaluate the integration of the first and second terms in the left-hand side of Eq. (5.29) by substituting Eqs. (5.30) and (5.31) into Eq. (5.29):

5.2

Kinematics of Interface

151

( j · n)ζ =ζ e dS e − ( j · n)ζ =ζ i dS i Se Si ∂( j · n) 1 1 = h ( j · n)ζ =0 dS + h dS + O(h 2 ) + R R ∂ζ ζ =0 S S ∂( j · n) dS + O(h 2 ), =h (5.32) ( j · n)ζ =0 (∇ · n) + ∂ζ ζ =0 S

where the change of the integration domains is obtained as follows. Suppose that (ξ, η) is a system of orthogonal coordinates on the interface, and (ξ e , ηe ) and (ξ i , ηi ) are similar coordinates on the surface S e and S i , respectively. Hence we have dS e = dξ e dηe = (1 + ζ e /R)(1 + ζ e /R )dξ dη, dS i = dξ i dηi = (1 + ζ i /R)(1 + ζ i /R )dξ dη,

(5.33) (5.34)

and then neglecting higher orders of h, with the help of Eq. (5.14), leads to Eq. (5.29), i.e., . ∂( j · n) ( j · n)(∇ · n) + j · nC dl + O(h 2 ), dS + h ∂ζ SV S C (5.35) where |ζ =0 is dropped. Notice that the right-hand side of Eq. (5.28) can be rewritten as

j · NdS V = h

∇ · j dV = h V

∇ · j dS,

(5.36)

S

and the vector field j can be decomposed as j = j ⊥ + j ,

(5.37)

where j ⊥ and j are the components of j normal and parallel to the interface, respectively. Equating the right-hand sides of Eqs. (5.36) and (5.35), dividing through by h, and taking the limit h → 0, we obtain .

j · n dl = C

C

S

∇ S · j dS,

(5.38)

where ∇ S · j denotes the surface divergence of j . With this definition of the surface divergence, Eq. (5.38) is the analogue of the Gauss divergence theorem in three-dimensional space applied to the interface. Using Eq. (5.38), we can rewrite Eq. (5.26) by substituting vs into j :

152

5 Dynamics of Spherical Vapor Bubble

1 DΞ = ∇ · vs − (n · ∇)(vs · n) = (v s · n)(∇ · n) + ∇ S · vs . Ξ Dt

(5.39)

Finally, the surface divergence of a vector field vs0 which is constant in time can be obtained from Eq. (5.39) as ∇ S · vs0 = −(vs0 · n)(∇ · n).

(5.40)

5.2.5 Equilibrium Thermodynamics of the Interface We derive an expression for entropy of the interface by a simple argument based on Carnot’s cycles [11, 19]. It should be noted that we restrict our discussion only to a reversible process. Consider a system composed of two bulk fluids facing each other with an interface of area A between. Let this area increase reversibly by δ A, keeping the temperature T of the system constant. It should be noted that the interface considered here is that of either a drop or a bubble in a bulk fluid; hence, A can be increased by deforming this drop or bubble. We assume that the bulk fluids do no work during the surface increment, for simplicity. Next, we decrease the temperature of the system to T = T − dT , with dT > 0, keeping the area constant; then, we decrease the area by the same amount δ A keeping the temperature T constant. Finally, we increase the system temperature by dT , i.e., raise the temperature to the initial temperature T keeping the area A constant. These sequential processes complete a cycle. We can calculate an amount of work δW done by the interface during this cycle: δW = [−σ (T ) + σ (T )]δ A = −(dσ/dT )δ AdT.

(5.41)

Now, we consider heat transport during this cycle. Notice that the net amount of heat absorbed by the bulk fluids is zero, but the interface itself absorbs and emits the net amount of heat. We define this absorbed heat amount δ Q at the temperature T , and the emitted heat amount δ Q at the temperature T . Using both the first and the second laws of thermodynamics, we have δW = δ Q − δ Q and δ Q/T = δ Q /T = δS, where S is the surface entropy. These equations lead to δW = (1 − T /T )δ Q = (dT /T )δ Q.

(5.42)

Equating Eqs. (5.41) and (5.42), we thus obtain δS = −(dσ/dT )δ A.

(5.43)

Similarly, change δE in internal energy E of the interface is given by δE = T δS − δW ; hence, using Eq. (5.43), we have δE = (σ − T (dσ/dT )) δ A.

(5.44)

5.3

General Conservation Equation at Interface

153

Hence, the entropy per unit area SS = δS/δ A and the energy per unit area e S = δE/δ A are obtained [13]. Finally, Helmholtz free energy per unit area f S of the interface is obtained as f S = e S − T SS = Fs /δ A = σ.

(5.45)

It should be reminded that the total amount of work performed by the system during the isothermal transformation is given by −δF. Equation (5.45) is consistent with this fact.

5.3 General Conservation Equation at Interface 5.3.1 Conservation Equations in Bulk Fluids The bulk fluid that is not on the interface is either a pure liquid or a pure vapor; hence we consider here conservation equations for a single-phase single-component pure fluid. As we have already defined in Sect. 5.2.3, the superscript e denotes the quantities evaluated in the external side of the interface, and i does those in the internal side of the interface. When we do not have to specify if a quantity is in either side, we write the quantity without superscript (Fig. 5.4). The conservation equations of mass, momentum, and energy, in general, are respectively written as [12] ∂ρ + ∇ · (ρv) = 0, ∂t ∂ (ρv) + ∇ · (ρvv) = −∇ p + ∇ · τ + ρb, ∂t ∂ (ρe) + ∇ · (ρev) = − p(∇ · v) + ε : τ − ∇ · q + ρ S, ∂t

(5.46) (5.47) (5.48)

where ρ is the mass density of the fluid, v is the velocity of the fluid, e is the internal energy per unit mass, p is the pressure, q is the heat flux vector, b is the body force per unit mass, S is the heat generated per unit time and unit volume. The viscous stress tensor τ in Eqs. (5.47) and (5.48) is given by Interface

Fluide

i

Fluid

Fig. 5.4 Definition of quantities evaluated in the external side of the interface and in the internal side of the interface

154

5 Dynamics of Spherical Vapor Bubble

1 τi j = 2μ εi j − δi j εkk + μb εkk δi j 3

with εi j =

$ 1# vi, j + v j,i , 2

(5.49)

where μ is the shear viscosity, μb is the bulk viscosity, and here and hereafter, the summation convention [7] is used (see Appendices A and B at the end of this book). Carefully observing Eqs. (5.46), (5.47), and (5.48), we can find that these equations can be rewritten in a unified form as ∂ (ρ f ) = −∇ · (ρ f v + φ) + ρϑ. ∂t

(5.50)

We assume that Eq. (5.50) can apply to the interface itself with the expressions for the various quantities identical in form to those applicable in the bulk fluids. Based on this assumption, we find a certain relationship between normal components of the fluxes on the two sides of the interface.

5.3.2 Conservation Equation in Frame Moving with Interface Let ξ (t) be the instantaneous position of a point P on the interface in the laboratory frame F, and v 0 = ξ˙ be its velocity, where the overdot indicates the derivative with respect to time. Let us express an arbitrary point in F. We introduce a new reference frame F with coordinates (x , t ). The frame F is moving with P with axes parallel to those of F such that P is instantaneously at rest in F . The relation between the frames F and F are given as x = x − ξ ,

t = t.

(5.51)

With the use of the chain rule, the differential operators ∂/∂t and ∇ are expressed in terms of (x , t ) by ∂ ∂ = − v0 · ∇ , ∂t ∂t

∇ = ∇ .

(5.52)

In particular, the particle velocity v in F is expressed by the particle velocity v in F as v = v − v 0 .

(5.53)

Notice that v 0 is the velocity of the specified point P, and hence it is a function of t alone. Substituting Eq. (5.52) into Eq. (5.50) gives ∂ (ρ f ) = −∇ · (φ + ρ f v ) + ρϑ. ∂t

(5.54)

Equation (5.54) represents the conservation equation in the moving frame F .

5.3

General Conservation Equation at Interface

155

5.3.3 Integration Form of Conservation Equation Consider an infinitesimal volume element V in the form of cylinder orthogonal to the interface and protruding equally to interior and exterior regions at the interface. This volume element cuts an infinitesimal surface element S from the interface, as seen in Sect. 5.2.3. Let P be a point on S. Integration of Eq. (5.54) over the volume of the cylinder in the frame F moving with the interface can be carried out by noticing that the integration and differentiation are commutable since the infinitesimal volume element V can be regarded as constant in the moving frame: d dt

∇ · (φ + ρ f v )dV +

ρ f dV = − V

V

ρϑdV + Σ,

(5.55)

V

where the last term Σ accounts for a process corresponding to the contribution of a conserved quantity on or along the interface. The first integral in the left-hand side can be rewritten with the use of the Gauss divergence theorem as

∇ · (φ + ρ f v )dV = V

SV

(φ + ρ f v ) · NdS V ,

(5.56)

where S V denotes the closed surface of V and N is the unit normal directed outward at a point of S V . Substituting Eq. (5.56) into Eq. (5.55) leads to d dt

ρ f dV = − V

SV

(φ + ρ f v ) · NdS V +

ρϑdV + Σ.

(5.57)

V

In principle, defining s as the surface density of a conserved quantity, we have the following characteristics on s: (i) Accumulation of the conserved quantity on the interface at a rate −∂s/∂t per unit area and unit time. The minus sign is because an increase in s decreases the outgoing flux; (ii) Generation of s on the interface at a rate χ per unit area and unit time; (iii) Convective transport of s along the interface with surface flux svs where vs denotes the tangential component of the surface velocity, v s in F ; (iv) Transport of s along the interface by means of a surface flux j of non-convective origin. Considering the above characteristics of s, we obtain the following explicit expression for Σ: . ∂s − + χ dS − ( j + svs ) · nC dC Σ= ∂t S C ∂s − + χ − ∇ S · ( j + svs ) dS, = ∂t S

(5.58)

where C is the closed line enclosing the surface element S of the interface, ∇S is the surface divergence in the moving frame F , and nC is the unit normal vector to C.

156

5 Dynamics of Spherical Vapor Bubble

Notice that the surface divergence theorem (5.38) and the relation of the differential operators between the laboratory and moving frames (5.52) are used.

5.3.4 Flux Balance on Interface We come back to Eq. (5.57) and take the height h of the circular cylinder to be an infinitesimal of a higher order than the diameter of S. We consider the limit of h → 0, noticing that the integrand of volume integrals, i.e, the term in the left-hand side and the second term in the right-hand side, in Eq. (5.57) are bounded. Then the volume integration in Eq. (5.57) can be eliminated but only the surface integration survives. i.e., 0=−

SV

(φ + ρ f v ) · NdS V + Σ.

(5.59)

Substituting Eq. (5.58) into Eq. (5.59) leads to

SV

(φ + ρ f v ) · NdS = V

S

∂s − + χ − ∇ S · ( j + svs ) dS. ∂t

(5.60)

In the limit of h → 0, the surface integral domain S V in the left-hand side, consists of only S e and S i which are defined in Sect. 5.2.3. Noticing that the areas of S, S e , and S i are the same, and defining n as the unit normal on the interface in the direction of the external region, we can rewrite Eq. (5.60) as

( ' e e e e (φ + ρ f v ) · n dS − [(φ i + ρ i f i v i ) · n]dS S S ∂s − + χ − ∇ S · ( j + sv s ) dS. = ∂t S

(5.61)

Since S is arbitrary, we obtain the following relation at an arbitrary point P on the interface: ∂s (φ e + ρ e f e v e ) − (φ i + ρ i f i v i ) · n = − + χ − ∇ S · ( j + sv s ). (5.62) ∂t The surface divergence of the last term in the right-hand side of Eq. (5.62) can be rewritten as ∇ S · (sv s ) = ∇ S · (sv s ) = v s · ∇ S s + s∇ S · (vs − v0 ) ( ' s DΞ , = s ∇ S · vs + (vs · n)(∇ · n) = Ξ Dt

(5.63)

5.3

General Conservation Equation at Interface

157

where Eqs. (5.39), (5.40), and (5.53) are used. It should be noted that v s vanishes at the point P by definition, while v s is in general nonzero at points in the neighborhood of P on the interface; hence, v0 = v s − v s = v s at the point P. Because s is defined only on the interface so that ∇s = ∇ S s, Eqs. (5.27) and (5.52) give ∂s = ∂t

∂ ∂ ∂ Ds + v0 · ∇ s = + vs · ∇ s = + vs · ∇ S s = . (5.64) ∂t ∂t ∂t Dt

Therefore, Eq. (5.62) is rewritten as (φ e + ρ e f e v e ) − (φ i + ρ i f i v i ) · n =−

( ' Ds − s −(vs · n)(∇ · n) + ∇ S · vs − ∇ S · j + χ . Dt

(5.65)

This equation shows that the outgoing normal flux of a conserved quantity Q equals to the incoming normal flux: (i) Minus the total time variation of the surface density of Q (the first term in the right-hand side); (ii) Minus the amount of Q necessary to maintain at the level s which is the surface density of Q in the newly formed surface area (the second term); (iii) Plus the non-convective influx of Q from neighboring points on the interface (the third term); (vi) Plus the rate of production of Q on the surface (the fourth term).

5.3.5 Conservation of Mass on Interface We now consider conservation equations on the interface between the pure liquid and the pure vapor by applying the general conservation equation, Eq. (5.65), on the interface. We first study the conservation equation of mass. We have already obtained the equation of conservation of mass in bulk fluid, Eq. (5.46): ∂ρ + ∇ · (ρv) = 0. ∂t This equation has the general form of the conservation equation (5.50): ∂ (ρ f ) = −∇ · (ρ f v + φ) + ρϑ. ∂t by taking f = 1, ϑ = 0, and φ = 0. Notice that the surface discontinuity can be defined in such a way that no molecules can accumulate on it [19]; hence, s = 0, j = 0, and χ = 0, then Eq. (5.65) reduces to ! " # $ ρ e v e − vs · n = ρ i v i − vs · n.

(5.66)

158

5 Dynamics of Spherical Vapor Bubble

With the use of Eq. (5.66), we can rewrite Eq. (5.65) as ' ( (φ e − φ i ) · n + ρ e ( f e − f i ) (v e − vs ) · n = (φ e − φ i ) · n + ρ i ( f e − f i ) (vi − vs ) · n =−

( ' Ds − s −vs · n(∇ · n) + ∇ S · vs − ∇ S · j + χ . Dt

(5.67)

Equation (5.66) simply expresses the mass flux balance on the interface: outgoing mass flux from one side of the interface should be equal to incoming mass flux to another side of the interface. It should be strongly emphasized that this equation provides no information regarding to the amount of mass flux and this amount must be evaluated at the molecular level according to the results of MD, molecular gas dynamics, and shocktube experiments presented in Chaps. 2 and 3. As discussed in Sects. 2.5 and 2.6 the physically correct boundary conditions for the velocities v V and temperature TV of the vapor at the interface to a Navier-Stokes set of equations are given by αe p∞ − p ∗ = ∗ p αc

√ 1 − αc (vV i − vwi ) n i α e − αc ∗ + , C4 − 2 π √ αc αc 2Rc TL TV − TL (vV i − vwi ) n i = d4∗ √ , TL 2Rc TL

(5.68) (5.69)

where TL is the liquid temperature at the interface, vw is the interface velocity, p∗ is the saturated vapor pressure, p∞ is the vapor pressure outside Knudsen layer, Rc is the gas constant per unit mass, αe is the evaporation coefficient, αc is the condensation coefficient, C4∗ and d4∗ are given in Chap. 2; αe and αc are given in Chap. 3. From the above two equations and the equation of state ( p∞ = ρ Rc TV ), we can obtain the mass flux j = ρ(vV i − vwi )n i as follows: p∞ / p∗ ! √ " j= ∗ c αe 2 π 1−α − C 4 αc

2 Rc

p∗ p∞ αe √ − αc √ , TL TL

(5.70)

√ where terms of O[(V / 2π TL )2 ] are dismissed. For αe = αc = α, Eq. (5.70) becomes p∞ / p ∗

j = √ 1−α 2 π α − C4∗

2 Rc

p∞ p∗ √ −√ TL TL

.

(5.71)

5.3

General Conservation Equation at Interface

159

5.3.6 Conservation of Momentum on Interface The conservation equation of momentum in bulk fluid is given by Eq. (5.47): ∂ (ρv) + ∇ · (ρvv) = −∇ p + ∇ · τ + ρb. ∂t As we have already observed in Sect. 5.3.1 that Eq. (5.47) can be rewritten in the general form of the conservation equation (5.50), ∂ (ρ f ) = −∇ · (ρ f v + φ) + ρϑ, ∂t by f = v, ϑ = b, and φ = p I − τ where I is the three-dimensional identity tensor. No momentum can accumulate on the interface so that s = 0, since the interface has zero mass. We can also take the surface source χ to vanish, except in rather special circumstances (e.g., an electrically charged interface). On the other hand, the surface momentum flux exists and is connected to the existence of surface tension. Notice that flux j in Eq. (5.67) to be considered here in this subsection has the form of the second order tensor. For f = v in Eq. (5.65), ρvv · n in the left-hand side of this equation is the momentum flux, i.e., force acting on a unit surface, so is ∇ S · j ; hence the total force acting on the surface S as defined in Fig. 5.3 with a closed contour C which is the line around S: C = ∂ S, on the interface is given by . . σ I S · nC dl, (5.72) F = − σ nC dl = ∇ S · j dS = C

S

C

where σ is the surface tension, nC is the outward unit normal to C, and I S is the two-dimensional identity tensor on the tangent plane of the interface. Comparing Eq. (5.72) with the explicit form of the conserved quantity taking place on or along the interface, Eq. (5.58), gives the expression of j : j = −σ I S

or jin = −σ δin .

(5.73)

If the positive direction on the closed contour C is chosen as the fingers of the right hand when the thumb is in the direction of the normal n to the surface S, the outward unit normal nC to the closed contour C is related to the unit tangent t C to ∂ S by nC = t C × n

C or n C n = kmn tk n m .

For further consideration of Eq. (5.72), the Stokes theorem [22] is useful: . u i tiC dl = n i i jk ∂ j u k dS, C

S

(5.74)

(5.75)

160

5 Dynamics of Spherical Vapor Bubble

where u i is an arbitrary vector. The right-hand side of Eq. (5.75) is an integration over the surface S, and the left-hand side is an integration over the closed contour C bounding S; hence the Stokes theorem is used to simplify line integrals. We take S as the surface S as defined in Fig. 5.3, and C as the closed contour which is the line around S. It turns the line integral of u along its boundary C of the closed surface S into the surface integral of the derivative of u (the curl) over the interior of S Simple expansion of Eq. (5.75) by replacing vector u i to any order of tensorial quantity (T . . .k ) provides the following generalized Stokes theorem [5] (see Appendix C at the end of this book)

. C

T . . .k tkC dl

=

n i i jk ∂ j (T . . .k )dS.

(5.76)

S

Notice that the integrand of the most right-hand side of Eq. (5.72) can be rewritten by the use of Eqs. (5.73) and (5.74) as C σ I S · nC dl = σ δln n C n dl = σ δln kmn n m tk dl.

(5.77)

Substituting (T . . .k ) = (σ δln kmn n m tk ) into Eq. (5.76), and applying Eq. (A.22) in Appendix A, we can rewrite Eq. (5.72) as .

Fl = C

(σ δln kmn n m )tkC dl =

n i i jk ∂ j (σ δln kmn n m )dS S

[n m ∂l (σ n m ) − n l ∂m (σ n m )] dS.

=

(5.78)

S

Comparing the integrand of Eqs. (5.72) and (5.78), and observing that n · n = 1 and that n and ∇α = ∇ S α are mutually orthogonal, we obtain ∇ S · j = −(σ/2)∇(n · n) + n(n · ∇σ ) + nσ (∇ · n) − ∇σ = nσ (∇ · n) − ∇σ.

(5.79)

With the use of Eqs. (5.47) and (5.79), we obtain ρ e v e (v e − vs ) · n + p e n − τ e · n = ρ i v i (v i − vs ) · n + pi n − τ i · n + ∇σ − nσ (∇ · n),

(5.80)

or, with the use of Eq. (5.67) we obtain the following form of conservation equation of momentum: ' ( ρ e (v e − v i ) (v e − vs ) · n = −( p e − pi )n + (τ e − τ i ) · n + ∇σ − nσ (∇ · n).

(5.81)

5.3

General Conservation Equation at Interface

161

5.3.7 Conservation of Energy on Interface Now, we consider the conservation equation of energy at the interface. The conservation equation of energy in bulk fluid is given by Eq. (5.48), i.e., ρ

∂e + (v · ∇)e + ∇ · q = − p(∇ · v) + ε : τ + ρ S. ∂t

(5.82)

We should recall that Eq. (5.48) can be rewritten in the general form of the conservation equation (5.50) ∂ (ρ f ) = −∇ · (ρ f v + φ) + ρϑ, ∂t with f = (1/2)v · v + e, ϑ = S, and the energy flux vector φ: φ = v · ( p I − τ ) + q,

(5.83)

substituted. In the local thermodynamic equilibrium, the internal energy per unit area e S of the interface, as obtained in Sect. 5.2.5, is written as e S = σ − T (dσ/dT ).

(5.84)

Substituting f = (1/2)v · v + e in Eq. (5.65), we obtain the term ρv · n that is the energy flux, in the left-hand side Eq. (5.65); hence j is the surface energy flux of non-convective origin. This flux j can be split into two terms accounting for the transport of mechanical energy and that of thermal energy. We consider the nonconvective transport of thermal energy across an element of area normal to the interface. We may assume that this transport vanishes as the element of area shrinks to a line lying on the interface. Thus j consists of only transport of mechanical energy. Now, we consider first the rate at which the mechanical work done by the surface tension force P on the surface S, as defined in Fig. 5.3, with the closed contour C on the interface as written by . P= σ v · nC dC. (5.85) C

Comparing this equation with Eqs. (5.54) and (5.58), we obtain j = −σ vs .

(5.86)

We may take the rate of surface energy production χ in Eq. (5.65) to vanish, as we have neglected the momentum surface source in our consideration on the conservation of the momentum. Substituting Eqs. (5.83), (5.84), and (5.86) into Eq. (5.67) leads to

162

5 Dynamics of Spherical Vapor Bubble

( ' 1 2 2 ρ e (v e − vs ) · n ee − ei + (v e − vi ) 2 # " ! $ + v e · ( pe I − τ e ) + q e − v i · ( pi I − τ i ) + q i · n ( D dσ dσ ' =T v S · n(∇ · n) + ∇ S · vs − σ −T Dt dT dT +v s · ∇ S σ + σ ∇s · vs .

(5.87)

5.4 Spherical Vapor Bubble From this section, we consider a more practical situation, e.g., bubble. We assume that the center of a bubble is fixed at a point in the space and only its volume changes. Hereafter flow parameters of liquid (in Re ) will be denoted by subscript L, and the corresponding parameters of vapor (in Ri ) by the subscript V . In general, we all know most small bubbles in ordinary liquids such as water are spherical; hence we assume that the bubble is spherical [23]. Hereafter we refer to the interface as the bubble wall and assume that flow fields are spherically symmetric. In Sect. 5.1, we have discussed that the interface at a time t is given by the set of points x satisfying Eq. (5.1): Ξ (x, t) = 0, where Ξ is a scalar function of time t and position x. We assume every physical quantity depends on the space coordinates through r only, the radial distance from the bubble center. Then we can express the interface as Ξ (r, t) = 0.

(5.88)

The equation for the bubble wall, i.e., Eq. (5.88), can be given explicitly as Ξ (r ) = r − R(t) = 0,

(5.89)

where R(t) is the instantaneous radius of the bubble. From Eqs. (5.9), (5.13), and (5.14), we obtain n= where r is the radial vector.

r , r2

∇·n=

2 , r

(5.90)

5.4

Spherical Vapor Bubble

163

5.4.1 Governing Equations for Spherical Bubble Now, we consider the governing equations for bulk liquid phase and bulk vapor phase [9], without indicating the subscript L and V . The discussion in the present subsection applies to the both phases. Using vector analysis formulas [4], differential operations are simplified as ∇·v =

∂v 2v + , ∂r r

∇s =

∂s r . ∂r r

(5.91)

The conservation equations of mass, Eq. (5.46), momentum, Eq. (5.47), and energy, Eq. (5.48), have been already obtained in Sect. 5.3.1. They are rewritten in slightly different forms as ∂ρ + ∇ · (ρv) = 0, ∂t ∂v + (v · ∇) v = −∇ p + ∇ · τ + ρb, ρ ∂t ∂e ρ + (v · ∇) e = − p(∇ · v) + ε : τ − ∇ · q + ρ S. ∂t

(5.92) (5.93) (5.94)

With the use of Eq. (5.91), the conservation equation of mass, Eq. (5.92), is rewritten as ∂ρ ∂ 2ρv + (ρv) + = 0. ∂t ∂r r

(5.95)

The conservation equation of momentum, Eq. (5.93), is greatly simplified in the spherical coordinates. Remind that ∇ × (∇ × v) is rewritten as i jk jlm vl,mk , then we obtain the following vector analysis relation: ∇ × (∇ × v) = ∇(∇ · v) − Δv.

(5.96)

Observing the fact that ∇ × v = 0 for spherically symmetric vector field v, we obtain Δv = ∇(∇ · v).

(5.97)

Tr(ε) = ∇ · v, 1 ∇ · ε = [∇(∇ · v) + Δv] . 2

(5.98)

Notice that

(5.99)

Then, with the use of Eq. (5.97), the radial component of the divergence of the second order tensor τ defined by Eq. (5.49) is given by

164

5 Dynamics of Spherical Vapor Bubble

(∇ · τ )r =

4μ + μb 3

∂ ∂r

∂v 2v + ∂r r

.

(5.100)

Substituting Eq. (5.100) into Eq. (5.93), we can obtain the conservation equation of the radial component of momentum: ∂v 1 ∂v +v = ∂t ∂r ρ

4μ + μb 3

∂ ∂r

∂v 2v + ∂r r

+ br −

1 ∂p , ρ ∂r

(5.101)

where v is the radial component of fluid velocity and br is the radial component of the external force. Now, we consider the conservation equation of energy. The second term in the right-hand side of Eq. (5.94) is the scalar product of two tensors: 1 ε : τ = 2μ εi2j − θ 2 + μb θ εi j δi j , 3

(5.102)

where θ is the divergence of velocity (∇ · v), and εi j is written as ⎛ ∂v

⎞ 0 0 ⎜ ∂r ⎟ v ⎟ ⎜ εi j = ⎜ 0 0 ⎟, ⎝ r v⎠ 0 0 r

(5.103)

for the spherically symmetric flow, and hence (εi j )2 =

∂v ∂r

2 +2

! v "2 r

.

(5.104)

Equation (5.102) thus becomes 4μ ε:τ = 3

∂v v − ∂r r

2 + μb

∂v 2v + ∂r r

2 .

(5.105)

Substituting Eq. (5.105) into Eq. (5.94), with Eq. (5.91), we obtain the conservation equation of energy: ρ

∂e ∂e +v ∂t ∂r

∂v 2v 4μ ∂v + − + ∂r r 3 ∂r 2v 2 ∂q 2q ∂v + − +μb − ∂r r ∂r r

= −p

where the heat flux vector q has the radial component q only.

v r

2

+ ρ S,

(5.106)

5.4

Spherical Vapor Bubble

165

5.4.2 Simplification For the ease of the discussion, we introduce further several simplifications in the following. We assume: (i) No body force is present, b = 0, (ii) Bulk viscosity coefficients of liquid and vapor are zero, μb = 0, (iii) The effect of the interaction between compressibility and viscosity is negligible since the liquid is essentially incompressible, (iv) Fourier’s Law, q = −λ∇T,

(5.107)

holds in both the liquid and vapor phases, where λ is the coefficient of the thermal conductivity. We assume that λ is constant for the liquid and λ is a function of temperature for the vapor, (v) No heat generation is present, S = 0. The normal components of velocities are rewritten as (v L · n)r =R(t) = w L ,

(v V · n)r =R(t) = wV ,

˙ (vs · n)r =R(t) = R,

(5.108)

where w L and wV are the liquid and vapor velocities at the bubble wall, respectively, and R˙ is the bubble wall velocity. 5.4.2.1 Conservation Equations for Bubble Exterior For the liquid in the exterior of the bubble, the conservation equation of mass, Eq. (5.95), is written as ρL ∂ ! 2 " ∂ρ L ∂ρ L + vL + 2 r v L = 0. ∂t ∂r r ∂r

(5.109)

Let us assume that the liquid is incompressible, ρ L = const., then the above equation can be further simplified: 1 ∂ ! 2 " r v L = 0. r 2 ∂r

(5.110)

Then, with the use of Eq. (5.108), the velocity is easily obtained: vL (r, t) =

R(t)2 w L (t). r2

(5.111)

The conservation equation of momentum is given by Eq. (5.101): 4μ L ∂ Dρ L 1 ∂ pL ∂v L ∂v L + vL = μbL + . + br − ∂t ∂r 3 ∂r Dt ρ L ∂r

(5.112)

The second term in the right-hand side is the body force term can be neglected by the assumption (i); b = 0. The first term in the right-hand side is apparently the

166

5 Dynamics of Spherical Vapor Bubble

interaction between compressibility (Dρ/Dt) and viscosity; hence this term can be neglected by the assumption (iii). Consequently, the following equation of motion ∂ pL = −ρ L ∂r

∂v L ∂v L + vL ∂t ∂r

,

(5.113)

is obtained. Substituting Eq. (5.111) into Eq. (5.113) and integrating with respect to r from r = ∞ to r , we obtain p L (r, t) − p L (r = ∞, t) = ρ L

˙ L + R 2 w˙ L 2R Rw 1 R 4 w2L − r 2 r4

.

(5.114)

Similarly, by neglecting the bulk viscosity and heat source [the assumption (ii)] and by substituting Eq. (5.107) [the assumption (iv)], we can simplify conservation equation of energy (5.106) as follows: ρL

∂e L ∂e L + vL ∂t ∂r

4μ L = 3

∂v L vL − ∂r r

2

λL ∂ + 2 r ∂r

2 ∂ TL r , ∂r

(5.115)

where we have used Eq. (5.110) to eliminate the first term in the right-hand side of Eq. (5.106). We assume that e L can be written as follows: e L = c L TL ,

(5.116)

where c L is the specific heat of liquid and assumed constant. We also assume that viscous heat dissipation, the second term in the right-hand side of Eq. (5.115), is negligible compared to heat conduction, the third term in the right-hand side of Eq. (5.115). With the use of Eq. (5.111), we obtain R 2 w L ∂ TL λL 1 ∂ ∂ TL + = ∂t ρ L c L r 2 ∂r r 2 ∂r

∂ TL r2 . ∂r

(5.117)

5.4.2.2 Conservation Equations for Bubble Interior The conservation equations of mass (5.95), momentum (5.101), and energy (5.106) for the vapor are respectively rewritten as " ∂ρV 1 ∂ ! 2 + 2 r vV ρV = 0, ∂t r ∂r ∂vV 1 ∂ pV 4 μV ∂ ∂vV + vV =− + ∂t ∂r ρV ∂r 3 ρV ∂r

∂vV 2vV + ∂r r

(5.118) ,

(5.119)

5.4

Spherical Vapor Bubble

ρV

∂eV ∂eV + vV ∂t ∂r

167

∂vV 2vV 4μV ∂vV vV 2 = − pV + + − ∂r r 3 ∂r r ∂ T 1 ∂ V + 2 r 2 λV , (5.120) r ∂r ∂r

where the assumptions (i)–(v) except for (iii) have been used. We assume that eV and pV can be written as follows: # $ eV = cvV (TV )TV = c pV (TV )/γ0 TV , pV = ρ V Rc TV ,

(5.121) (5.122)

where Rc is the gas constant per unit mass, cvV is the specific heat of vapor at constant volume of vapor. The ratio of specific heats, γ0 in Eq. (5.121), is defined by γ0 = cpV (Tr )/cvV (Tr ) = c pV 0 /cvV 0 ,

(5.123)

where Tr is a reference temperature, i.e., constant. Mayer’s relation is written as Rc = cpV0 − cvV 0 = cvV (Tr )(γ0 − 1),

(5.124)

and hence the equation of state (5.122) becomes pV =

γ0 − 1 c p0 ρV TV . γ0

(5.125)

We rewrite the first term in the right-hand side of Eq. (5.120) with the use of Eqs. (5.118), (5.122), and (5.124): 2vV ∂vV + − pV ∂r r ∂ pV ∂ pV ∂ TV ∂ TV = + vV + vV − ρV (c pV − cvV ) . ∂t ∂r ∂t ∂r

(5.126)

We also assume that viscous heat dissipation, the second term in the right-hand side of Eq. (5.120), is negligible compared to heat conduction, the third term in the right-hand side of Eq. (5.120). Then, substituting Eq. (5.126) back into Eq. (5.120) leads to ρV

∂(c pV TV ) ∂(c pV TV ) ∂ TV 1 ∂ d pV + vV + 2 = r 2 λV . (5.127) ∂t ∂r dt ∂r r ∂r

168

5 Dynamics of Spherical Vapor Bubble

5.4.3 Boundary Conditions With the use of Eq. (5.108), the conservation equation of mass at the bubble wall (5.66) becomes ˙ = ρ Lw (w L − R), ˙ ρV w (wV − R)

(5.128)

where ρV w and ρ Lw are densities of vapor and liquid at the bubble wall, respectively; however, we have already assumed that ρ L , μ L , and λ L are constant. Then ρ Lw = ρ L . The sign of Eq. (5.128) is positive for condensation and negative for evaporation. The molecular mass flux j [Eq. (5.70)] is defined to be positive for evaporation (negative for condensation). Therefore, Eq. (5.128) should be equal to − j. We discuss the above relation taking bubble expansion as an example. Suppose that a vapor bubble in an equilibrium state begins to expand by a decrease of the pressure in the ambient liquid, as shown in Fig. 5.5. As bubble grows, pressure inside bubble may decrease below saturated vapor pressure at the temperature of the liquid; hence evaporation occurs. This evaporation induces the difference between the vapor velocity at the interface and bubble wall velocity by an additional velocity associated with the mass flux, as shown in Fig. 5.5. Therefore we obtain ˙ = ρV w (wV − R) ˙ = − j. ρ L (w L − R)

(5.129)

From the above equation, the boundary condition of the bubble interior is given by wV = R˙ −

j , ρV w

Fig. 5.5 Velocity of bubble wall, and velocities of vapor and liquid at the bubble wall

(5.130)

5.4

Spherical Vapor Bubble

169

and that of the bubble exterior by j , w L = R˙ − ρL

(5.131)

where we can assume that the liquid density ρ at the bubble wall is the same as that in bulk fluid. We also have already known that the relative normal velocity of the vapor to the moving bubble wall is given by Eq. (5.70); hence Eqs. (5.130) and (5.131) can be explicitly written as p∞ / p ∗ ! √ " wV = R˙ − ∗ c αe ρV w 2 π 1−α αc − C 4 p∞ / p ∗ ! √ " w L = R˙ − ∗ c αe ρ Lw 2 π 1−α αc − C 4

2 Rc

p∗ p∞ αe √ , (5.132) − αc √ TLw TLw

2 Rc

p∗ p∞ αe √ , (5.133) − αc √ TLw TLw

where p ∗ is the saturated vapor pressure, and in the case of bubble, p ∗ is given by Kelvin equation [20]: ∗

p =

∗ p∞ exp

−

2σ ρ L R c TL R

,

(5.134)

∗ is the saturated vapor pressure for the plain interface, and written as a where p∞ function of the liquid temperature at the bubble wall only: ∗ ∗ = p∞ p∞ (TLw ) .

(5.135)

Now, we consider the conservation equation of momentum at the bubble wall. The general equation (5.81) can be rewritten as ρ Lw (v L − v V ) [(v L − vs ) · n] = −( p Lw − pV w )n + (τ Lw − τ V w ) · n + ∇σ − nσ (∇ · n).

(5.136)

With the use of Eq. (5.129), Eq. (5.81) is also rewritten as ˙ ρV w (wV − R)(w V − w L ) = − j (wV − w L ) 4μ L ∂v L v L = −( p L − pV w ) + − 3 ∂r r =R + r r =R + 2σ vV 4μV w ∂vV − − , − − 3 ∂r r =R − r r =R R or with the further use of Eqs. (5.130) and (5.131),

(5.137)

170

5 Dynamics of Spherical Vapor Bubble

ρV w 2σ j2 1− p Lw + = pV w + R ρV w ρL 4μ L ∂v L 4μV w ∂vV vL vV + − , (5.138) − − 3 ∂r r =R + r r =R + 3 ∂r r =R − r r =R − where R + is the radius of the bubble wall in the liquid and R − is that in the vapor. The conservation equation of energy (5.87): 1 2 2 ρ [(v L − vs ) · n] e Lw − eV w + (v L − v V ) 2 '# $ # $( + v L · ( p Lw I − τ Lw ) + q Lw − v V · ( pV w I − τ V w ) + q V w · n ( dσ ' D dσ vs · n(∇ · n) + ∇ S · vs − σ −T =T Dt dT dT (5.139) + vs · ∇ S σ + σ ∇ S · vs , is rewritten as 1 2 2 − j e Lw − eV w + (w L − wV ) 2 4μ L ∂v L v L + w L p Lw − − 3 ∂r r =R + r r =R + 4μV ∂vV vV = wV pV w − − 3 ∂r r =R − r r =R − ∂ TL ∂ TV 2σ ˙ + λL − λV w − R, ∂r r =R + ∂r r =R − R

(5.140)

where we assume that Fourier’s law (5.107) holds in both the liquid and vapor phases [the assumption (iv)], and that only σ vs · n(∇ · n) survives in the right-hand side of Eq. (5.139). It should be noted that there exists an ambiguity2 in definition of T appearing in Eq. (5.139); however, this ambiguity vanishes in the present study with the use of Eq. (5.66), and we obtain ∂ TL ∂ TV − λ Vw ∂r r =R + ∂r r =R − 2 2 1 1 j j = j L− + 2 ρL 2 ρV w

λL

2 The ambiguity is a result of the difference between the vapor temperature and liquid temperature at the interface, as shown in Fig. 5.5 [see also Eq. (5.69)].

5.5

Practical Description of Bubble Motion

171

∂v L 4μV w ∂vV v L vV − − − ∂r r =R + r r =R + 3ρV w ∂r r =R − r r =R − 2σ ˙ R, (5.141) + w L p Lw − wV pV w + R +

4μ L 3ρ L

where L is the latent heat: L = eV w +

pV w p Lw − e Lw − . ρV w ρL

(5.142)

Noticing that the latent heat term usually dominates over the other terms in the right-hand side of Eq. (5.141), we obtain ∂ TL ∂ TV − λV w = j L. λL ∂r r =R + ∂r r =R −

(5.143)

It should be reminded that there exists a temperature discontinuity at the bubble wall as already given by Eq. (5.69): TV w wV − R˙ = 1 + d4∗ √ . TLw 2Rc TLw

(5.144)

This discontinuity is also illustrated in Fig. 5.5.

5.5 Practical Description of Bubble Motion We now apply the formulations obtained in Sect. 5.4 to the growth and collapse of a vapor bubble. A bubble motion may start typically when the pressure in liquid increases or decreases, or when a bubble is placed in a super heated or sub cooled liquid. In the former case, the inertial effect may become the most dominant factor for the bubble motion, and on the other hand, in the later case the thermal effect may be prevailing. In the following, we derive the most general form of the solution which is applicable to both the inertially and thermally controlled bubble motion; however, the explicit form of the solution, in general, can only be obtained with the use of numerical analysis mainly because equations whose solutions should satisfy are integro-differential equations. We have derived all equations and conditions necessary for the analysis of the motion of a spherical bubble. Now, we further investigate the structure of the equation set.

172

5 Dynamics of Spherical Vapor Bubble

5.5.1 Flow Fields in Liquid First, we consider flow fields outside of a bubble, i.e., in liquid phase. Velocity field v L (r, t) and pressure field p L (r, t) are given by Eqs. (5.111) and (5.114), once R and w L and time derivatives of them are obtained. w L , in turn, can be obtained from Eqs. (5.130) and (5.131) with Eq. (5.70), with the use of the assumption that density ρ L is constant. Therefore with R obtained, v L and p L can be obtained. Now, we turn our attention to R. We derive a governing equation of bubble dynamics: ρL

3 R w˙ L + w 2L 2

2σ + 2 jw L = pV (r = R − , t) − p L (r → ∞, t) − R 2 ρV w 4μ L ∂v L vL j 1− + − + ρV ρL 3 ∂r r =R + r r =R + 4μV w ∂vV vV − , (5.145) − 3 ∂r r =R − r r =R −

where we have substituted Eq. (5.138) into Eq. (5.114) with r = R, with the use Eq. (5.131). Once the bubble wall velocity R˙ is obtained from Eq. (5.145), velocity fields inside and outside of the bubble can be determined from Eqs. (5.70), (5.132), and (5.133). Then, the conservation equation of energy inside the bubble, Eq. (5.127), should be solved to carry on our discussion further.

5.5.2 Uniform Pressure in Bubble Interior First, we assume that the pressure inside the bubble is uniform, and a function of time only: pV = pV (t).

(5.146)

The reason why this assumption is appropriate is discussed in the latter half of this subsection. Then, Eq. (5.145) can be further simplified by assuming that both the normal viscous stress and density in vapor phase are negligible compared to those in liquid phase: ρL

3 R w˙ L + w 2L 2

+ 2 jw L = pV (t) − p∞ (t) −

2σ j2 wL , (5.147) − 4μ L + R R ρV

where p∞ = p L (r → ∞, t), and Eq. (5.111) is used. Notice that neglecting ˙ Then, the molecular mass flux j [Eq. (5.70)] in Eq. (5.131), we obtain w L = R. Eq. (5.147) becomes the well-known Rayleigh–Plesset equation [9].

5.5

Practical Description of Bubble Motion

173

We now verify the legitimacy of Eq. (5.146). It should be emphasized here that we do not suppose violent bubble motions such as rapid bubble collapse caused by a strong shock wave in the liquid, but rather mild bubble motions such that the bubble wall velocity can be assumed well below the speed of sound in the vapor. Recall that the local pressure disturbances in the bubble propagate with the speed of sound a in the vapor. If a is significantly larger than the characteristic speed of bubble motion, i.e., the speed of radial oscillation, VR , Eq. (5.146) can hold. Here, we consider the characteristic time tC , which can be defined by the characteristic length scale divided by the characteristic velocity. We use a bubble radius R0 as the characteristic length. Then, the characteristic time for pressure propagation tC p can be simply determined: tC p = R0 /a.

(5.148)

The characteristic time for bubble motion can be estimated as follows. First, we can assume the adiabatic process in the bubble caused by a moderately large VR , where the heat exchange across the bubble wall may not be completed during the characteristic time for bubble motion. We want to deduce such a smallest characteristic time for the discussion of appropriateness of Eq. (5.146). With the use of Eq. (5.121), the increment of internal energy deV can be written as deV = cvV 0 dTV ,

(5.149)

where cvV in Eq. (5.121) is assumed constant for simplicity. Total derivative of Eq. (5.125) becomes pV γ0 − 1 d pV − 2 d pV = c pV 0 dTV . ρV γ0 ρV

(5.150)

Substituting Eq. (5.149) and an adiabatic relation deV = −( pV /ρV2 )dρV into Eq. (5.150), we obtain γ0

dρV d pV = , ρV pV

(5.151)

where Mayer’s relation (5.124) is used. Integrating Eq. (5.151) leads to the following adiabatic relation: pV γ = const. ρV0

(5.152)

With the use of Eq. (5.152) and neglecting viscous effects, the characteristic time tCm that is defined as the inverse of the natural frequency of bubble oscillation is given by [15]

174

5 Dynamics of Spherical Vapor Bubble Table 5.1 Thermodynamic properties of saturated water at 300 K ρV 0 c pV γ0 a σ λV kg/m3 kJ/(kg·K) m/s mN/m mW/(m·K)

pV 0 kPa

ρL

3.534

996.62

0.02556

1.872

1.332

428.8

71.69

18.47

DV mm2 /s 386.0

Table 5.2 Characteristic times associated with transport phenomena in bubble motion (µs) tC p tCm tC T 2.33

648

tC m

pV 0 2σ = 2π 3γ0 − 2 ρ L R0 ρ L R03

1660

−1/2 ,

(5.153)

where pV 0 and R0 are undisturbed pressure and radius, respectively. For further discussion, it is recommended to refer to [9]. We also estimate the characteristic time of propagation of heat √ conduction, tC T . With the use of the combination of variables, we have χ = r/(2 DV t), where DV is the coefficient of thermal diffusivity and defined by DV = λV /(ρV c pV ),

(5.154)

(see Appendix C at the end of this book). Then, we can obtain the expression of tC T as follows: tC T = R02 /(4DV ).

(5.155)

Now, we can evaluate the characteristic times with the use of thermodynamic properties of vapor and water at the saturated condition at 300 K from a saturated water table [10], which are listed on Table 5.1. We suppose R0 is 1 mm, then we can calculate tC p , tCm , and tC T with the use of Eqs. (5.148), (5.153), and (5.155), respectively. The results are shown in Table 5.2. We found that tC p is significantly smaller than the other characteristic times associated with bubble motion and transport phenomena in a bubble; hence, the propagation of pressure disturbance is much faster than the bubble motion itself. This result supports the assumption of the uniform pressure inside bubble. In contrast, tC T is relatively large; then, we should take the temperature distribution for the bubble interior into consideration.

5.5.3 Temperature, Pressure, and Velocity Fields With the use of Eq. (5.146), the temperature field in vapor can be obtained by solving conservation equation of energy in vapor, Eq. (5.127). Solution inside the bubble,

5.5

Practical Description of Bubble Motion

175

i.e., temperature field in the vapor TV , is decomposed into two parts: isentropic temperature field Tis and perturbation temperature field Θ. The former is a function of time only and the latter is a function of both time and space. It should be emphasized here that a pair of temperature boundary conditions, Eqs. (5.143) and (5.144), is not a priori specified but it is a part of the solution. Finally, pressure pV (t) can be obtained from the conservation equation of mass, Eq. (5.118), the conservation equation of energy, Eq. (5.127), and the equation of state, Eq. (5.125), with the use of the fact that p V is independent of the space as shown in Eq. (5.146). Then, the velocity distribution in the bubble, vV , can also be obtained.

5.5.4 Boundary Conditions of Temperature Field We can obtain the temperature field in the liquid, R(t) < r < ∞, and that in the vapor, 0 < r < R(t), by solving Eqs. (5.117) and (5.127), respectively, with proper boundary conditions. The boundary conditions at infinity in the liquid and that on the center of the vapor bubble, respectively, are lim TL (r, t) = TL0 ,

r →∞

lim

r →0

∂ TV (r, t) (r, t) = 0, ∂r

(5.156)

(5.157)

where TL0 is the temperature of undisturbed liquid at the bubble wall in liquid. Notice that Eqs. (5.117) and (5.127) contain the second-order derivatives in space and the first-order derivatives in term. Therefore, the number of boundary conditions in space for each equation should be two, and consequently four boundary conditions are necessary. With the use of Eq. (5.71), the other two conditions are Eqs. (5.143) and (5.144): ∂ TL = λV w λL ∂r r =R + ∂ TV = λV w + ∂r r =R −

TV w

∂ TV + jL ∂r r =R − ∗ 2 p p∞ / p ∗ p∞ L , (5.158) − √ √ √ 1−α TL TL 2 π α − C 4∗ Rc

˙ ∗ wV − R , = TLw 1 + d4 √ 2Rc TLw

(5.159)

where R + and R − are the radius of the bubble wall in the liquid and the radius of the bubble wall in the vapor, respectively.

176

5 Dynamics of Spherical Vapor Bubble

If the boundary conditions for the temperature at the bubble wall had been explicitly specified, we should have solved the Dirichlet problem. If the boundary conditions for the temperature gradient at the bubble wall had been explicitly specified, we should have solved the Neumann problem. However, the boundary conditions for the temperature and temperature gradient at the bubble wall are coupled through Eqs. (5.158) and (5.159); hence we have to solve the mixed boundary value problem. For the practical sake of solving the problem, we consider that Eq. (5.158) defines the boundary condition for the temperature gradient at the bubble wall in the liquid and that Eq. (5.159) defines the boundary condition for the temperature at the bubble wall in the vapor. Therefore we solve the Neumann problem for the temperature field in the liquid and the Dirichlet problem for the temperature field in the vapor.

5.6 Temperature Field of Bubble Exterior As we discussed in Sect. 5.5.4, with the use of Eqs. (5.156) and (5.158), the temperature field in the liquid, R(t) < r < ∞, can be determined with the following boundary and initial conditions: # $ ∂ TL (r = R + , t) = TLw = λV w TV w + j L /λ L , ∂r lim TL (r, t) = TL0 , r →∞

TL (r, t = 0) = TL0 ,

(5.160) (5.161) (5.162)

is the where the prime denotes the partial differentiation with respect to r and TLw temperature gradient in the direction of r at the bubble wall in the liquid, and R + is the radius of the bubble wall in the liquid. It should be emphasized here that . The temperature at the bubble wall in the liquid Eq. (5.160) is the definition of TLw is denoted by TLw :

TL (r = R + , t) = TLw .

(5.163)

5.6.1 Lagrangian Formulation One of difficulties lying in studying thermo-fluid dynamic problems with interface is that the interface moves. This class of problems is categorized, in general, as “moving boundary problems”. It is often useful to study equations in Lagrangian coordinates for one-dimensional moving boundary problems. We introduce Lagrangian coordinates (z L , τ ) which are related to Eulerian coordinates (r, t) by the following relations [9]: z L (r, t) =

r R(t)

ξ 2 dξ, τ = t.

(5.164)

5.6

Temperature Field of Bubble Exterior

177

Now, we consider the time derivative with z L fixed. With the use of Eq. (5.164), we obtain dz L =

∂z L ∂z L ˙ dr + dt = r 2 dr − R(t)2 R(t)dt = 0. ∂r ∂t

(5.165)

From Eq. (5.164), we have 1 ∂ ∂ = 2 . ∂z L r ∂r

(5.166)

2 dr R ∂r ˙ R(t), = = ∂t z L dt r

(5.167)

Equation (5.165) leads to

where |z L is the operation with z L fixed, d/dt is the derivative with respect to t with z L fixed. Then, we take the derivative with respect to τ : 2 ∂ ∂ ∂r R ∂ ∂ ∂ ˙ R(t) = + = + , ∂τ z L ∂t r ∂t z L ∂r t ∂t r r ∂r t

(5.168)

where Eq. (5.167) is used. With the use of Eqs. (5.130) and (5.168), the left-hand side of Eq. (5.117) becomes R 2 w L ∂ TL ∂ TL j R 2 ∂ TL ∂ TL + = − , ∂t r r2 ∂r t ∂τ z L ρ L r 2 ∂r t

(5.169)

and then, with the use of Eqs. (5.166) and (5.169), Eq. (5.117) is rewritten as j R 2 ∂ TL ∂ TL λL ∂ 4 ∂ TL − r (z L , τ ) . = ∂τ ρ L ∂z L ρ L c L ∂z L ∂z L

(5.170)

In the following, the subscript L is dropped for simplicity.

5.6.2 Transformation of Variables Now we replace an independent variables τ and z with new variables s and η defined, respectively, by

178

5 Dynamics of Spherical Vapor Bubble

τ

s=

R 4 (ξ )dξ,

(5.171)

0

z . η= √ (λ/(ρc)

(5.172)

Then, Eq. (5.170) can be written as ∂T ∂T ∂ ! r "4 ∂ T + vcL (s) = , ∂s ∂η ∂η R ∂η

(5.173)

where vcL (s) is defined by vcL (s) = −

j ρ R2

ρc , λ

(5.174)

and ∂s/∂τ = R(τ )4 is used. Subtracting ∂ 2 T /∂η2 from the both sides of Eq. (5.173), we have ∂T ∂ ∂2T ∂T = + vcL (s) − 2 ∂s ∂η ∂η ∂η

∂T r4 −1 , R4 ∂η

(5.175)

with boundary and initial conditions: 1 ∂ T (η = 0, s) = TLw,η = 2 ∂η R

$ 1 # λV w TV w + j L , λρc

(5.176)

lim T (η, s) = TL0 ,

(5.177)

T (η, 0) = TL0 .

(5.178)

η→∞

We note that T (η = 0, s) = TLw .

(5.179)

Notice now that ∂ T /∂η and ∂ 2 T /∂η2 should be small enough to neglect the righthand side of Eq. (5.175), except in the neighborhood of r = R, while (r 4 /R 4 − 1) should be extremely small there; hence Eq. (5.175) reduces to the heat equation with fixed boundary: ∂T ∂T ∂2T . + vcL (s) = ∂s ∂η ∂η2

(5.180)

5.6

Temperature Field of Bubble Exterior

179

5.6.3 Laplace Transform of Heat Equation Equation (5.180) with conditions (5.179)–(5.178) can be solved using Laplace transform. We use the two-sided Laplace transform (bilateral Laplace transform) [24]: ϕ(ξ, s) = Lη {φ(η, s)} =

∞

−∞

e−ηξ φ(η, s)dη.

(5.181)

It should be noted that the lower limit of the integration in Eq. (5.181) is −∞. In contrast, the lower limit of the integration of one-sided Laplace transform, which is commonly referred to as the Laplace transform, is 0. The usefulness of using the two-sided Laplace transform will be revealed in Sect. 5.6.4.2. We notice that the domain of definition of T is 0 T < ∞; hence we need to extend the domain of definition of T to −∞ < T < ∞ to apply Eq. (5.181). We define T E whose the domain of definition is −∞ < T < ∞ and 0 s < ∞. We impose the following on T E and derivatives of T E with respect to η: for η > 0, T E (η, s) = T (η, s) ∂ T E (η, s) = ∂ T (η, s)/∂η for η > 0, ∂η ∂ 2 T E (η, s) = ∂ 2 T (η, s)/∂η2 for η > 0, ∂η2

(5.182) (5.183) (5.184)

and lim T E (η, s) = T (0, s),

(5.185)

lim ∂ T E (η, s)/∂η = ∂ T (0, s)/∂η,

(5.186)

lim ∂ 2 T E (η, s)/∂η2 = ∂ 2 T (0, s)/∂η2 .

(5.187)

η→+0 η→+0 η→+0

We also impose the restriction on T E at s = 0 as % T E (η, s = 0) =

T (η, s = 0) for η > 0, 0 for η < 0.

(5.188)

We consider the heat conduction equation (5.180). Noticing the derivatives with respect to η when η < 0 are physically undetermined, we rewrite the heat conduction equation for T E as ∂T E ∂2T E ∂T E , + H (η)vcL (s) = H (η) ∂s ∂η ∂η2 where H (η) is the Heaviside function:

(5.189)

180

5 Dynamics of Spherical Vapor Bubble

% H (η) =

1 for η > 0, 0 for η < 0.

(5.190)

Then, the two-sided Laplace transform of T E (η, s) can be properly defined as 5 6 ϕ(ξ, s) = Lη T E (η, s) =

∞

−∞

e−ηξ T E (η, s)dη =

∞

e−ηξ T (η, s)dη.

0

(5.191)

Similarly, each term in Eq. (5.189) can be transformed as % Lη

∂T E ∂s

& =

∂ϕ , ∂s

(5.192)

% & ∂T E = vcL (s) [−T (η = 0, s) + ξ ϕ(ξ, s)] , Lη vcL (s) ∂η % Lη

∂2T E ∂η2

& =−

(5.193)

∂T (η = 0, s) + ξ [−T (η = 0, s) + ξ ϕ(ξ, s)] . ∂η

(5.194)

Then, the Laplace transform of Eq. (5.189) becomes ∂ϕ (ξ, s) − vcL (s)T (η = 0, s) + ξ vcL (s)ϕ(ξ, s) ∂s ∂T =− (η = 0, s) − ξ T (η = 0, s) + ξ 2 ϕ(ξ, s). ∂η

(5.195)

With the use of Eqs. (5.160), (5.163), and (5.176), Eq. (5.195) can be rewritten as ∂ϕ + ξ(vcL − ξ )ϕ = TLw (vcL − ξ ) − TLw,η . ∂s

(5.196)

The method of variation of parameters [2] with Eqs. (5.178), (5.182), and (5.185) gives the solution of Eq. (5.196): s vcL (ζ )dζ ϕ(ξ, s) = ϕ0 (ξ ) exp ξ 2 s − ξ 0 s 8 7 + TLw (ς ) [vcL (ς ) − ξ ] − TLw,η (ς ) · exp ξ 2 (s − ς ) − ξ 0

ς

s

vcL dζ dς, (5.197)

5.6

Temperature Field of Bubble Exterior

181

where 5

6

∞

e−ηξ TL0 dη ϕ0 (ξ ) = Lη T (η, 0) = 0 ∞ = TL0 e−ηξ H (η)dη = TL0 Lη {H (η)} , E

(5.198)

−∞

with the definition of T E (η, 0): T E (η, 0) = TL0 H (η).

(5.199)

5.6.4 Inverse Laplace Transform of Heat Equation We evaluate inverse Laplace transform of ϕ(ξ, s) in Eq. (5.197), by calculating the inverse Laplace transform of the following four functions: (i) ϕ1 (ξ, s) = exp(ξ 2 s).

(5.200)

(ii) ϕ10 (ξ, s) = ϕ0 (ξ ) exp ξ 2 s − ξ

s

vcL (ζ )dζ .

(5.201)

0

(iii)

s

ϕ20 (ξ, s) =

'

( TLw (ς )vcL (ς ) − TLw,η (ς ) exp ξ 2 (s − ς ) − ξ

ς

0

s

vcL dζ dς. (5.202)

(iv)

s

ϕ30 (ξ, s) = −

TLw (ς )ξ exp ξ 2 (s − ς ) − ξ

0

s ς

vcL dζ dς.

(5.203)

Then, the inverse Laplace transform of ϕ(ξ, s) in Eq. (5.197) is given by −1 φ(η, s) = L−1 η {ϕ(ξ, s)} = Lη {ϕ10 (ξ, s) + ϕ20 (ξ, s) + ϕ30 (ξ, s)} .

(5.204)

5.6.4.1 Inverse Laplace Transform of ϕ1 (ξ, s) First, we evaluate the inverse Laplace transform of ϕ1 (ξ, s), which can be calculated as follows:

182

5 Dynamics of Spherical Vapor Bubble Im(z) Re(z)

+A

–A O – ip – A

– ip

– ip + A

Fig. 5.6 The integral path used in evaluation of the inverse Laplace transform of eξ

2s

σ +i∞ 5 2 6 1 2 ξ s e = φ1 (η, s) = = eξ s · eξ η dξ 2πi σ −i∞ 2 −i p+∞ " ! 1 η (5.205) exp −z 2 dz, = √ exp − 4s 2π s −i p−∞ L−1 η {ϕ1 (ξ, s)}

L−1 η

where σ is a real number [24], and we substituted ξ = 2π λi with the use of the following definitions: √ η p=σ s+ √ . 2 s

√ iη z = 2π sλ − √ , 2 s

(5.206)

Now, we evaluate the integration in Eq. (5.205) with the use of the integral path shown in Fig. 5.6. The integration can be rewritten as lim

−i p+A

A→∞ −i p−A

exp(−z 2 )dz = lim

A→∞

−A

−i p−A

+

A −A

+

−i p+A

exp(−z 2 )dz.

A

(5.207) The first and third integrations in Eq. (5.207) vanishes since exp(−z 2 ) is a rapidly decaying function as |z| → ∞; hence we obtain 2 1 η φ1 (η, s) = √ exp − . 4s 2 πs

(5.208)

5.6.4.2 Laplace Transform of Distribution In order to proceed further to evaluate the inverse Laplace transform of ϕ(ξ, s) in Eq. (5.197), we need the Laplace transform of distribution. The derivative of the Heaviside function in the sense of distribution [21] is given by dH (η) = δ(η), dη

(5.209)

where δ(η) is the Dirac delta function, or delta distribution, which satisfies for an arbitrary integrable and continuous function f (η),

5.6

Temperature Field of Bubble Exterior

∞

183

f (η)δ(η)dη = f (0).

−∞

(5.210)

Then, the two-sided Laplace transform of δ(η) is carried out with the use of integration by parts: Lη {δ(η)} =

∞ −∞

e−ηξ δ(η)dη =

∞

−∞

e−ηξ

dH (η) dη = ξ dη

∞

e−ηξ dη = 1,(5.211)

0

where Eqs. (5.190) and (5.209) are used. Similarly, 5 6 (1) Lη δ (η) =

∞

∞ dδ(η) (η)dη = e−ηξ dη dη −∞ −∞ ∞ ' (∞ = e−ηξ δ(η) −∞ − (−ξ )e−ηξ δ(η)dη = ξ.

e

−ηξ (1)

δ

−∞

(5.212)

Since we used the two-sided, not one-sided, Laplace transform, we can escape from the evaluation of δ(η = 0) in Eq. (5.212) and H (η = 0) in Eq. (5.211). This is the main reason why we choose to use the two-sided Laplace transform. The following inverse Laplace transforms are useful: L−1 η {1} = δ(η),

(5.213)

(1) L−1 η {ξ } = δ (η).

(5.214)

5.6.4.3 Inverse Laplace Transform of ϕ10 (ξ, s) Next, we evaluate the inverse Laplace transform of ϕ10 (ξ, s) in Eq. (5.201). By the convolution theorem [24], φ10 (η, s) can be written as φ10 (η, s) = L−1 η {ϕ10 (ξ, s)} = TL0 H (η) ∗ φ11 (η, s),

(5.215)

where ∗ is the convolution, and the following definition of φ11 (η, s) is used: % −1 2 {ϕ φ11 (η, s) = L−1 exp ξ (ξ, s)} = L s − ξ 11 η η

s

& vcL (ζ )dζ

.

(5.216)

0

With the use of Eqs. (5.205) and (5.208), Eq. (5.216) can be transformed to yield 9 2 : s 1 1 vcL (ζ )dζ η− . φ11 (η, s) = √ exp − 4s 2 πs 0

(5.217)

By substituting Eqs. (5.204) and (5.217) into Eq. (5.215), and carrying out convolution, φ10 (η, s) is obtained:

184

5 Dynamics of Spherical Vapor Bubble

φ10 (η, s) = TL0

∞

0

9 2 : s 1 1 dς. (5.218) vcL (ζ )dζ − ς η− √ exp − 4s 2 πs 0

With the use of the following definition: v=

η−

*s 0

vcL (ζ )dζ − ς , √ 2 s

1 dv = − √ dς, 2 s

(5.219)

we obtain the following expression of φ10 (η, s)3 : TL0 φ10 (η, s) = 2

9

1 + erf

η−

*s

vcL (ζ )dζ √ 2 s

0

: .

(5.220)

5.6.4.4 Inverse Laplace Transform of ϕ20 (ξ, s) We, then, evaluate the inverse Laplace transform of ϕ20 (ξ, s) in Eq. (5.202) as follows: φ20 (η, s) = L−1 η {ϕ20 (ξ, s)} s ( ' 1 TLw (ς )vcL (ς ) − TLw,η (ς ) = √ 2 π 0 ⎡ ! "2 ⎤ *s η − v (ζ )dζ ς cL 1 ⎢ ⎥ exp ⎣− ×√ ⎦ dς, s−ς 4(s − ς )

(5.221)

where we have used that the integration and inverse Laplace transform are commutable. 5.6.4.5 Inverse Laplace Transform of ϕ30 (ξ, s) Finally, we evaluate the inverse Laplace transform of ϕ30 (ξ, s) in Eq. (5.203). φ30 (η, s) can be written as φ30 (η, s) = L−1 η {ϕ30 (ξ, s)} % s 2 = L−1 ξ T (ς ) exp ξ (s − ς ) − ξ − Lw η 0

ς

s

& vcL (ζ )dζ dς . (5.222)

The (a.k.a. the Gauss error function) erf(x) is defined as erf(x) = * x error function 2 0 exp(−ξ )dξ , and the complementary error function erfc(x) is defined as erfc(x) = 1 − √ *∞ erf(x) = √2π x exp(−ξ 2 )dξ . It should be noted that limx→∞ erf(x) = √2π · 2π = 1.

3

√2 π

5.6

Temperature Field of Bubble Exterior

185

Now, we consider the inverse Laplace transform of the following function: % −1 2 {ϕ φ31 (η, s) = L−1 ξ exp ξ (ξ, s)} = L (s − ς ) − ξ 31 η η

s

ς

& vcL (ζ )dζ

.

⎛ 2 ⎞ *s η − v (ζ )dζ ς cL dδ(η) 1 ⎜ ⎟ exp ⎝− = ∗ √ ⎠ dη 4(s − ς ) 2 π(s − ς ) =

∞ −∞

⎛ 2 ⎞ *s (η − y) − v (ζ )dζ cL ς dδ(y) 1 ⎟ ⎜ exp ⎝− √ ⎠ dy dy 2 π(s − ς ) 4(s − ς )

1 =− √ 2 π

η−

*s ς

vcL (ζ )dζ

2(s − ς )3/2

⎛ 2 ⎞ *s η − ς vcL (ζ )dζ ⎜ ⎟ exp ⎝− ⎠, 4(s − ς )

(5.223)

where Eqs. (5.212) and (5.217), and the following inverse Laplace transform is used: % 2 L−1 exp ξ (s − ς ) − ξ η

s

ς

⎛ 1 ⎜ exp ⎝− = √ 2 π(s − ς )

& vcL (ζ )dζ

η−

*s ς

vcL (ζ )dζ

2 ⎞

4(s − ς )

⎟ ⎠.

(5.224)

Substituting Eq. (5.223) into Eq. (5.222), noticing that the integration and inverse Laplace transform are commutable, we obtain 1 φ30 (η, s) = √ 2 π

0

s

TLw (ς )

η−

*s ς

vcL (ζ )dζ

2(s − ς )3/2

⎛ 2 ⎞ *s η − ς vcL (ζ )dζ ⎜ ⎟ × exp ⎝− ⎠ dς. 4(s − ς )

(5.225)

Putting subscript L back again in equations, and with the use of Eqs. (5.204), (5.220), (5.221), and (5.225), the temperature field is obtained:

186

5 Dynamics of Spherical Vapor Bubble

TL (η, s) = φ10 (η, s) + φ20 (η, s) + φ30 (η, s) *s η − 0 vcL (ζ )dζ TL0 1 + erf = √ 2 2 s 1 + √ 2 π 1 + √ 2 π

0

⎛

TLw (ς )vcL (ς ) − TLw,η (ς ) ⎜ exp ⎝− √ s−ς

s

s

TLw (ς )

0

η−

*s ς

vcL (ζ )dζ

2(s − ς )3/2

η−

*s ς

vcL (ζ )dζ

2 ⎞

4(s − ς )

⎟ ⎠ dς

⎛ 2 ⎞ *s η − ς vcL (ζ )dζ ⎜ ⎟ exp ⎝− ⎠ dς. 4(s − ς ) (5.226)

The integrals in the right-hand side of Eq. (5.226) involve both TLw and TLw,η . TLw,η that is defined by Eq. (5.176) has been provided as Neumann boundary condition, as we discussed in Sect. 5.5.4. In contrast, TLw has not been specified; hence we proceed to determine TLw in the next subsection.

5.6.5 Liquid Temperature at Bubble Wall Our next task is to determine liquid temperature at bubble wall, TLw , by taking the limit of Eq. (5.226): TLw (s) = lim T (η, s) = lim [φ10 (η, s) + φ20 (η, s) + φ30 (η, s)] . η→0

η→0

(5.227)

Evaluation of the limit of the first and second terms in the right-hand side of Eq. (5.227) is rather simple4 : TL0 lim φ10 (η, s) = η→0 2

9

* s 1 − erf

0

vcL (ζ )dζ √ 2 s

: ,

(5.228)

lim φ20 (η, s)

η→0

1 = √ 2 π

s 0

⎧ * 2 ⎫ s ⎪ ⎪ ⎨ ⎬ v (ζ )dζ ς cL TLw (ς )vcL (ς ) − TLw,η (ς ) exp − dς. √ ⎪ ⎪ s−ς 4(s − ς ) ⎩ ⎭ (5.229)

4

erf(−x) =

√2 π

* −x 0

exp(−ξ 2 )dξ = − √2π

*x 0

exp(−ζ 2 )dζ = −erf(x), where ζ = −ξ .

5.6

Temperature Field of Bubble Exterior

187

In contrast, the evaluation of the third term in Eq. (5.227) requires an extra caution in taking the limit of the following function: lim φ32 (η, s)

η→0

1 = lim √ η→0 2 π 1 = lim √ η→0 2 π

⎧ 2 ⎫ *s ⎪ ⎨ ⎬ s η − ς vcL (ζ )dζ ⎪ η − TLw (ς ) exp dς ⎪ ⎪ 2(s − ς )3/2 4(s − ς ) 0 ⎩ ⎭ 9 ' *s (2 : ∞ η − s−τ vcL (ζ )dζ TLw (s − τ ) exp − (2dσ ), η 4η2 /(4σ 2 )) √

2 s

(5.230) √ where τ = s − ς and σ = η/(2 τ ). Equation (5.230) leads to lim φ32 (η, s)

η→0

9 2 : ∞ s 1 1 dσ TLw (s − τ ) exp − σ − √ vcL (ζ )dζ = lim √ η η→0 π 2 τ s−τ √ 2 s TLw (s) ∞ 1 = √ exp(−σ 2 )dσ = TLw (s), (5.231) 2 π 0

where we have evaluated the limit of the second term in the square bracket in exponential function in the second row of Eq. (5.231) as follows: 1 √ η, τ →0 2 τ

s

lim

1 √ vcL (s) s − s + τ + O(τ 2 ) η, τ →0 2 τ vcL (s) √ = lim τ = 0. (5.232) η, τ →0 2

vcL (ζ )dζ = lim

s−τ

We, then, can evaluate the limit of the third term in the right-hand side of Eq. (5.227): 1 TLw (s) 2 ⎧ * 2 ⎫ *s s ⎪ ⎪ s ⎬ ⎨ v (ζ )dζ ς cL 1 ς vcL (ζ )dζ dς. (5.233) − TLw (ς ) exp − √ ⎪ ⎪ 4(s − ς ) 2(s − ς )3/2 2 π 0 ⎭ ⎩

lim φ30 (η, s) =

η→0

Substituting Eqs. (5.228), (5.229), and (5.233) into Eq. (5.227), we obtain

188

5 Dynamics of Spherical Vapor Bubble

TLw (s)

s TLw,η (ς ) 1 = TL0 − TL0 erf [Z L (s, 0)] − √ exp −Z L (s, ς )2 dς √ s−ς π 0 s 1 1 1 TLw (ς ) vcL (ς ) − vm L (s, ς ) √ +√ exp −Z L (s, ς )2 dς, 2 s−ς π 0 (5.234)

where *s vm L (s, ς ) =

ς

vcL (ζ )dζ s−ς

*s ,

Z L (s, ς ) =

vcL (ζ )dζ . √ 2 s−ς

ς

(5.235)

Solving above integral equation with the use of TLw,η defined by Eq. (5.176), we can obtain TLw . Then, we can properly determine T (η, s), by substituting TLw and TLw,η into Eq. (5.226).

5.6.6 Gradient of Liquid Temperature at Bubble Wall We further proceed to determine the gradient of liquid temperature at the bubble wall, TLw,η . The result obtained in this subsection will be useful in evaluating TV w,η in Sect. 5.7.4. First, we differentiate Eq. (5.226) with respect to η: ⎧ 2 ⎫ *s ⎨ η − 0 vcL (ζ )dζ ⎬ TL0 ∂ TL (η, s) = √ exp − √ ⎭ ⎩ ∂η 2 πs 2 s *s s ( η − ς vcL (ζ )dζ ' 1 − √ TLw (ς )vcL (ς ) − TLw,η (ς ) 2(s − ς )3/2 2 π 0 ⎧ ⎫ 2 * ⎪ ⎨ η − ςs vcL (ζ )dζ ⎪ ⎬ × exp − dς ⎪ ⎪ 4(s − ς ) ⎩ ⎭ ⎧ 2 ⎫ *s ⎪ s ⎨ ⎬ η − ς vcL (ζ )dζ ⎪ 1 1 TLw (ς ) − + √ 3/2 ⎪ ⎪ 2 π 0 4(s − ς )5/2 ⎩ 2(s − ς ) ⎭ ⎧ ⎫ 2 * ⎪ ⎨ η − ςs vcL (ζ )dζ ⎪ ⎬ × exp − dς, (5.236) ⎪ ⎪ 4(s − ς ) ⎩ ⎭ and then, we need to take the limit of Eq. (5.236) as η → 0. With the use of Eq. (5.233), the limit of Eq. (5.236) becomes,

5.7

Temperature Field of Bubble Interior

189

∂ TL (η, s) η→0 ∂η 2TL0 = √ exp −Z L (s, 0)2 − TLw (s)vcL (s) πs s ∂ 7T (ς ) exp '−Z (s, ς )2 (8 Lw L 1 ∂ς −√ dς √ s−ς π 0 s ' ( vm (s, ς ) 1 TLw (ς )vcL (ς ) − TLw,η (ς ) √ exp −Z L (s, ς )2 dς + √ 2 π 0 (s − ς ) s vm (s, ς )2 1 (5.237) TLw (ς ) √ exp −Z L (s, ς )2 dς. − √ 2 s−ς 2 π 0

TLw,η (s) = lim

Substituting Eq. (5.234) into the second term of the right-hand side of Eq. (5.237), we finally obtain 2TL0 TLw,η (s) = √ exp −Z L (s, 0)2 − vcL (s)TL0 {1 − erf [Z L (s, 0)]} πs s ∂ 7T (ς ) exp '−Z (s, ς )2 (8 Lw L 1 ∂ς dς −√ √ s−ς π 0 & s% 1 1 + √ TLw (ς ) vcL (ς ) − vm L (ς ) − TLw,η (ς ) 2 2 π 0 vm L (s, ς ) − vcL (s) (5.238) · exp −Z L (s, ς )2 dς. √ (s − ς )

5.7 Temperature Field of Bubble Interior Now, we turn our attention to temperature field in vapor. As we discussed in Sect. 5.5.4, with the use of Eqs. (5.157) and (5.159), the temperature field in vapor in the region 0 < r < R(t), can be determined with the following boundary and initial conditions: ˙ − ∗ wV − R , (5.239) TV (r = R , t) = TV w = TLw 1 + d4 √ 2Rc TLw ∂ TV (r, t) (r, t) = 0, (5.240) lim r →0 ∂r (5.241) TV (r, t = 0) = TV 0 , where TV 0 is the temperature of undisturbed vapor, TV w is the temperature of vapor at the bubble wall, the prime denotes the partial differentiation with respect to r , and R − is the radius of the bubble wall in the vapor. It should be emphasized here that Eq. (5.239) is the definition of TV w . The temperature gradient in the direction of r at the bubble wall in the vapor is denoted by TV w :

190

5 Dynamics of Spherical Vapor Bubble

∂ TV (r = R + , t) = TV w . ∂r

(5.242)

In the following, the subscript V is dropped for simplicity. Then, Eq. (5.127) becomes ρ

∂(c pv T ) ∂(c pv T ) dp 1 ∂ ∂T +v = + 2 λr 2 . ∂t ∂r dt ∂r r ∂r

(5.243)

5.7.1 Adiabatic Solution We restrict the problem considered to the case that thermal boundary layer is relatively thin compared with the bubble radius. Then, the temperature field outside the thermal boundary layer is, with the neglect of the thermal conduction term in Eq. (5.243), given by ρ

$ dp d # = 0. cpT − dt dt

(5.244)

Substituting Eq. (5.125), we can rewrite Eq. (5.244) as $ γ0 − 1 1 d p d # . cpT = c p0 T dt γ0 p dt 1

(5.245)

It is known [14] that for an ideal gas, c p = c p0 νcp T νc ,

(5.246)

may hold for wide ranges of temperature and pressure, where νc is a constant. Substituting Eq. (5.246) into Eq. (5.245) leads to γ0 − 1 T d p d νc +1 )− (T = 0. dt νcp γ0 p dt

(5.247)

Imposing the condition that p = pV 0 at T = TV 0 , the integration of Eq. (5.245) with Eq. (5.246) leads to Tis = TVνc0 +

γ0 − 1 νc ln νcp (νc + 1) γ0

p pV 0

1/νc

.

(5.248)

It should be noted that Tis in Eq. (5.248) is a solution of Eq. (5.245), not that of (5.243); hence Tis cannot satisfy the boundary conditions (5.239) and (5.242).

5.7

Temperature Field of Bubble Interior

191

5.7.2 Lagrangian Formulation As we have already used in Sect. 5.6.1, here, we also introduce the Lagrangian coordinates (z, τ ): z(r, t) =

R(t)

ρ(ξ, t)ξ 2 dξ,

τ = t.

(5.249)

r

We should notice the difference of Eq. (5.249) from Eq. (5.164) in the integration domain and ρ in Eq. (5.249). With the use of Eq. (5.249), the total derivative is given by dz =

∂z ∂z ˙ dr + dt = −ρr 2 dr + ρ R(t)2 R(t)dt = 0, ∂r ∂t

(5.250)

for fixed z. Then, we obtain 2 R dr ∂r ˙ = R(t), = ∂t z dt r j ∂ T R 2 W ∂ T ∂ T R2 ˙ ∂ T + 2 = + 2 R− ∂t r r ∂r t ∂t r r ρ ∂r t 2 ∂ T jR ∂ T = − . ∂τ z ρr 2 ∂r t

(5.251)

(5.252)

However, we should notice that 1 ∂ ∂ =− 2 , ∂z ρr ∂r

(5.253)

which is derived from Eq. (5.249), causes the significant difference in the governing equation. With the use of Eqs. (5.253) and (5.252), Eq. (5.243) is transformed to yield ∂(c p T ) 1 dp ∂ 2 ∂(c p T ) 4 ∂T + jR = + λρr (z, τ ) . ∂τ ∂z ρ dτ ∂z ∂z

(5.254)

5.7.3 Boundary Layer Solution We assume that the difference between the temperature T inside the bubble and the isentropic temperature Tis which is given by Eq. (5.248) is small, i.e., T (r, t) = Tis (t) + Θ(r, t)

with || 1,

(5.255)

192

5 Dynamics of Spherical Vapor Bubble

and hence, physical properties, c p and λ, which are dependent on temperature, in Eq. (5.243) can be written as functions of Tis alone. With T replaced with Tis , Eq. (5.246) is rewritten as c p = c p0 νcp Tisνc .

(5.256)

We recall the well-established method in compressible boundary layer theory called the Howarth–Illingworth–Stewartsort transformation [14] in which it is assumed that μ is independent of the pressure, and the following relation holds: μ = cμ Tis ,

(5.257)

where cμ is a constant. Notice that Pr =

μc p = const. λ

(5.258)

is valid for wide ranges of temperature and pressure. From Eqs. (5.246), (5.257), and (5.258), we obtain λ = cλV Tisνc +1 ,

(5.259)

where cλV is a constant. Then, with the use of Eqs. (5.256) and (5.259), Eq. (5.254) can be transformed to yield νc ∂(Tisνc T ) γ0 − 1 T d p cλV ∂ 2 ∂(Tis T ) 4 νc +1 ∂ T + jR − = ρr Tis . (5.260) ∂τ ∂z νcp γ0 p dτ c p0 νcp ∂z ∂z Substituting Eq. (5.255) into Eq. (5.260) leads to ν +1 ∂(Tisc ) γ0 − 1 T d p − ∂τ νcp γ0 p dτ νc ∂(Tisνc Θ) ∂(Tisνc Θ) cλV ∂ 2 ∂(Tis Θ) 4 + jR − ρr Tis = 0. + ∂τ ∂z c p0 νcp ∂z ∂z (5.261)

We notice that terms in the first square bracket are exactly the same as the lefthand side of Eq. (5.247); hence these terms disappear. Thus, Eq. (5.247) is eventually a partial differential equation of Q with respect to τ and z as ∂Q cλV ∂ ∂Q + j R2 − ∂τ ∂z c p0 νcp ∂z

∂Q 4 ρr Tis = 0, ∂z

(5.262)

5.7

Temperature Field of Bubble Interior

193

where Q is defined by Q(r, t) = Tisνc (t)Θ(r, t).

(5.263)

5.7.4 Solution of Heat Equation Now, as in Sect. 5.6.2, we replace independent variables τ and z with new variables s and η defined respectively by

τ

p(ξ )R 4 (ξ )dξ, νcp (γ0 − 1) η = c p0 z, cλV γ0 s=

(5.264)

0

(5.265)

where we should notice that Eq. (5.264) is different from Eq. (5.171) in having p in the integrand. Then, Eq. (5.254) can be written as ∂ ! r "4 ∂ Q ∂Q ∂Q + vcV (s) = , ∂s ∂η ∂η R ∂η

(5.266)

where vcV is defined by vcV (s) = c p0

νcp (γ0 − 1) j . cλV γ0 p R2

(5.267)

We obtained Eq. (5.266) with the use of Eq. (5.125) and the followings: ∂s = p(τ )R(τ )4 , ∂τ ρTis γ0 γ0 ρTis = = (1 + O()). p c p0 (γ0 − 1) T c p0 (γ0 − 1)

(5.268)

(5.269)

The initial and boundary conditions for T , Eq (5.239) to Eq. (5.241), can be rewritten as T (η = 0, s) = TV w lim

η→∞

∂ T (η, s) = 0, ∂η T (η, 0) = TV 0 .

˙ ∗ wV − R , = TLw 1 + d4 √ 2Rc TLw

(5.270) (5.271) (5.272)

194

5 Dynamics of Spherical Vapor Bubble

The temperature gradient in the direction of η at η = 0 in the vapor is denoted by TV w,η : 1 1 ∂ T (η = 0, s) = TV w,η = 2 ∂η R c p0

cλV γ0 T . νcp (γ0 − 1) V w

(5.273)

Then, the initial and boundary conditions for Q are obtained as Q(η = 0, s) = Q w = Tisνc (TV w − Tis ), lim Q(η, s) = 0,

η→∞

Q(η, 0) = 0.

(5.274) (5.275) (5.276)

We note that ∂ Q(η = 0, s) = Q w,η = Tisνc TV w,η . ∂η

(5.277)

With the use of the same discussion in derivation of Eq. (5.180), Eq. (5.266) can be reduced to the heat equation with fixed boundary: ∂Q ∂Q ∂2 Q . + vcV (s) = ∂s ∂η ∂η2

(5.278)

We notice that Eq. (5.278) has the same form as that of Eq. (5.180), with vcV defined by Eq. (5.267) instead of vcL as advection velocity. With Eq. (5.226) and Eqs. (5.274), (5.275), and (5.276), the solution of Eq. (5.278) is obtained. Putting subscript V back again to variables, and then, using Eqs. (5.255) and (5.263), the temperature field inside the bubble is obtained: Q(η, s) Tisνc s [TV w (ς ) − Tis (ς )] vcV (ς ) − TV w,η (ς ) 1 = Tis + √ √ s−ς 2 π 0 ⎧ 2 ⎫ * s ⎪ ⎨ η − ς vcV (ζ )dζ ⎪ ⎬ × exp − dς ⎪ ⎪ 4(s − ς ) ⎩ ⎭ *s s η − ς vcV (ζ )dζ 1 [TV w (ς ) − Tis (ς )] + √ 2(s − ς )3/2 2 π 0 ⎫ ⎧ 2 * ⎪ ⎬ ⎨ η − ςs vcV (ζ )dζ ⎪ dς. (5.279) × exp − ⎪ ⎪ 4(s − ς ) ⎭ ⎩

TV (η, s) = Tis +

5.7

Temperature Field of Bubble Interior

195

With the use of Eqs. (5.234), (5.235), (5.274), and (5.277), TV at the bubble wall can be written as s TV w,η (ς ) 1 exp −Z (s, ς )2 dς TV w (s) = Tis − √ √ s−ς π 0 s vcV (ς ) − 12 vmV (s, ς ) 1 +√ [TV w (ς ) − Tis (ς )] √ s−ς π 0 × exp −Z V (s, ς )2 dς,

(5.280)

where *s vmV (s, ς ) =

ς

vcV (ζ )dζ s−ς

*s ,

Z V (s, ς ) =

vcV (ζ )dζ . √ 2 s−ς

ς

(5.281)

Similarly, the gradient of vapor temperature inside the bubble can be written as *s s ( η − ς vcV (ζ )dζ ' 1 ∂ TG (η, s) TV w (ς )vcV (ς ) − TV w,η (ς ) =− √ ∂η 2 π 0 2(s − ς )3/2 ⎧ ⎫ 2 * ⎪ ⎨ η − ςs vcV (ζ )dζ ⎪ ⎬ × exp − dς ⎪ ⎪ 4(s − ς ) ⎩ ⎭ ⎧ 2 ⎫ *s ⎪ ⎪ s ⎨ ⎬ η − v (ζ )dζ ς cV 1 1 TV w (ς ) − + √ 3/2 ⎪ ⎪ 2 π 0 4(s − ς )5/2 ⎩ 2(s − ς ) ⎭ ⎧ 2 ⎫ * ⎪ ⎨ η − ςs vcV (ζ )dζ ⎪ ⎬ × exp − dς. (5.282) ⎪ ⎪ 4(s − ς ) ⎩ ⎭ Then, with the use of Eq. (5.238), we obtain the gradient of the vapor temperature at the bubble wall, TV w,η , s ∂ 7[T (ς ) − T (ς )] exp '−Z (s, ς )2 (8 Vw is V 1 ∂ς TV w,η (s) = − √ dς √ s−ς π 0 & s% 1 1 [TV w (ς ) − Tis (ς )] vcV (ς ) − vmV (ς ) − TV w,η (ς ) + √ 2 2 π 0 5 6 vmV (s, ς ) − vcV (s) × exp −Z V (s, ς )2 dς. (5.283) √ (s − ς )

196

5 Dynamics of Spherical Vapor Bubble

Solving above integral equation with the use of TV w defined by Eq. (5.270), we can obtain TV w,η . Then, we can properly determine T (η, s), by substituting TV w and TV w,η into Eq. (5.279).

5.7.5 Pressure and Velocity In Sect. 5.7.3, we assumed that pressure pV inside the bubble is uniform and a function of time only, i.e., Eq. (5.146). Now, we determine how pV depends on time, and then, determine the explicit functional form of vV (r, t). Adding the conservation equation of mass, Eq. (5.118), multiplied by c pV TV and the conservation equation of energy, Eq. (5.127), leads to ∂ ∂ TV ∂ d pV (ρV c pV TV ) − (r 2 ρV vV c pV TV ) − r 2 λV = −r 2 . ∂r ∂r ∂t dt (5.284) With the use of Eqs. (5.125), (5.246), and (5.256), we obtain ρV c pV TV = ρV c pV 0 νcp Tisνc TV =

νcp γ0 νc T pV , γ0 − 1 is

(5.285)

which is a function of time only; hence the square bracket term in the right-hand side of Eq. (5.284) is independent of r . Integrating once the both sides of Eq. (5.284) with respect to r leads to

νcp γ0 ∂ TV vV Tisνc pV − λV γ0 − 1 ∂r

r3 =− 3

d dt

d pV − r . dt (5.286) An integration constant which should have appeared in Eq. (5.286) has been eliminated with the use of the condition at r = 0. Equation (5.286) leads to 2

νcp γ0 νc T pV γ0 − 1 is

γ0 − 1 d 3 γ0 − 1 Tis ∂ TV d νc ln(Tis pV ) − ln( pV ) = − vV − , dt r νcp γ0 pV ∂r νcp γ0 Tisνc dt

(5.287)

where Eq. (5.259) is used. Substituting Eq. (5.248) into Eq. (5.287), the left-hand side of Eq. (5.287) becomes d γ0 − 1 d d ln(Tisνc ) + ln( p V ) − ln( p V ) dt dt νcp γ0 Tisνc dt 1 d pV γ0 − 1 = 1− . νcp γ0 (νc + 1)Tisνc pV dt

(5.288)

Since pV is independent of r , we can impose condition at the bubble wall, r = R, on Eq. (5.288), and consequently, we obtain

5.8

Structure of Mathematical Model

197

; ∂ T γ0 − 1 γ0 − 1 1 − . Tis − p w V V νcp γ0 ∂r r =R − νcp γ0 (νc + 1)Tisνc (5.289) From Eqs. (5.287), (5.288), and (5.289), We obtain d pV 3 = dt R

r ∂ TV r Tis γ0 − 1 ∂ TV − (r, t) − (r = R , t) . vV (r, t) = wV + R pV νcp γ0 ∂r R ∂r

(5.290)

5.8 Structure of Mathematical Model We have derived a set of equations in Sects. 5.4, 5.5, 5.6, and 5.7, to deal with the dynamics of a spherical vapor bubble accompanied with evaporation and condensation at the bubble wall. We refer to this set of equations as a mathematical model that describes dynamics of a spherical vapor bubbles. Here, we investigate the structure of this model. We have constructed the model from conservation equations (5.110), (5.113), and (5.117) for liquid, and (5.118), (5.119), and (5.127) for vapor, with boundary conditions at the bubble wall, Eqs. (5.129), (5.138), (5.143), and (5.144). We should notice that these equations are partial differential equations. Although t and r are originally taken as a set of the independent variables in these equations, we changed these variables to s and η defined and used in Sects. 5.6.1, 5.6.2, 5.7.2, and 5.7.4 for ease of solving the equations. After these changes of variables, the Laplace transform, and other manipulations, we successfully obtained ordinary differential equations, integral equations, and integro-differential equations with auxiliary algebraic equations, instead of partial differential equations. In addition to the equations we have already obtained, the two equations

t

R(t) =

˙ )dτ + R(0), R(τ

(5.291)

w˙ L (τ )dτ,

(5.292)

0 t

w L (t) = 0

are necessary to close the set of equations. All equations that compose the mathematical model describing the dynamics of a spherical vapor bubble with the evaporation and condensation at the bubble wall are listed in Table 5.3 with variables in each equation. The types of equation, ordinary differential equation, integral equation, integro-differential equation, and algebraic equation, are also indicated as O.D.E, I.E., I.D.E., and A.E., respectively in Table 5.3. Here, we verify that the number of variables (24) is the same as the number of equations listed in Table 5.3 (24); hence the set of equations we have derived is successfully closed. It should be also emphasized that we do not have to solve

198

5 Dynamics of Spherical Vapor Bubble

Equation

Table 5.3 Variables and constants in necessary equations Type Contained variables Constants

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

(5.70) (5.114) (5.116) (5.121) (5.122) (5.130) (5.131) (5.134) (5.135) (5.142)

A.E. A.E. A.E. A.E. A.E. A.E. A.E. A.E. A.E. A.E.

(11)

(5.147)

O.D.E.

(12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22)

(5.160) (5.174) (5.234) (5.248) (5.239) (5.256) (5.257) (5.259) (5.267) (5.283) (5.289)

A.E. A.E. I.E. A.E. A.E. A.E. A.E. A.E. A.E. I.D.E. O.D.E.

(23) (24)

(5.291) (5.292)

A.E. A.E.

j, TLw , p ∗ ˙ w L , w˙ L p L , R, R, e L , TL eV w , c pV , TV w ρV w , p V , TV w ˙ j, ρV w wV , R, ˙ wL , j R, ∗ ,T , R p ∗ , p∞ Lw ∗ p∞ , TLw eV w , e Lw , p V w , p L , ρV w , L w˙ L , w L , R, j, pV , ρV w , w L , j, L, λ TV w , TLw Vw vcL , j, R ,v TLw , TLw cL Tis , p V ˙ wV TV w , TLw , R, c pV , Tis μV , Tis λV , Tis vcV , j, R, ρV w , pV w TV w , TV w , Tis , vcV pV , R, TV w , TV w , Tis , wV R, R˙ w L , w˙ L

αe , αc , C4∗ , Rc ρL cL γ0 Rc

Conditions p∞ p∞

ρL ρ L , Rc , σ ρL ρ L , σ , μL

p∞

λL ρL , λL , cL νc , νcp , γ0 d4∗ , Rc νc , νcp , c p0 cμV cλV , νc γ0 , cλV , c p0 , νcp

TL0 TV 0 , p V 0

γ0 , νc , νcp R(0)

partial differential equations. This greatly reduces the complexity of mathematical procedures. ˙ wV , In these twenty four variables, the following twenty three variables: R, R, , T , T , v , v , ∗ , ρ w L , w˙ L , j, pV , p ∗ , p∞ , e , e , L, T , T , T Vw Vw L Vw Lw is cV cL Vw Lw μV w , λV w , and c pV , are either directly or indirectly depend on t only. Only one variable, p L , is a function5 of both t and r ; however, we know that p L can be ˙ w L , and w˙ L are obtained for any given r , as written obtained explicitly once R, R, in Eq. (5.114). Sixteen constant physical properties required are the following: αe , αc 6 , C4∗ , ∗ d4 ,Rc ,γ0 , ρ L , σ , c L , λ L , μ L , c p0 , cλV , cμV , νcp , νc . Then, this set of equations

5

We notice that the independent variable s can be replaced back with t alone, with the use of Eqs. (5.171) and (5.264); hence, variables depending only on s can be rewritten as those depending only on t. We can also replace back independent variable η with both r and t, with the use of Eqs. (5.166), (5.172), (5.253), and (5.265). 6 As we discussed in Chap. 3, α is not a constant; however, we treat α as a constant in this chapter c c for simplicity.

5.9

Bubble Expansion with Uniform Interior

199

can be solved with initial and boundary conditions including p∞ , TL0 , TV 0 , and pV 0 , specified. It should be noted that we use the temperature boundary condition (5.69), listed in the nineteenth row in Table 5.3, with Eqs. (5.234) and (5.280) for TLw and TV w , respectively, and the temperature gradient boundary condition (5.143) which is listed in the twentieth row in Table 5.3, with Eqs. (5.238) and (5.283) for TLw and TV w , respectively. It should also be noted that both Eqs. (5.234) and (5.238), which are listed in the twelfth row in Table 5.3, are derived from the same equation (5.226); hence only one of them are used as an independent equation, and that both Eqs. (5.280) and (5.283), which in the eighteenth row in Table 5.3 are derived from the same equation (5.279). Thus, we have derived the set of equations, which can describe dynamics of a spherical vapor bubble by taking phase change at the bubble wall accurately into consideration. However, the derived set of equations is rather complicated; hence can be solved only numerically. Solving this set of equations is out of the scope of this book. Further discussion may be found in, e.g., Beylich [3]. Instead, we further simplify the mathematical model with some crucial assumptions in the next section.

5.9 Bubble Expansion with Uniform Interior We have derived a set of equations for the radial motion of a spherical vapor bubble with taking phase change at the vapor–liquid interface into account correctly in Sects. 5.4, 5.5, 5.6, 5.7, and 5.8. In this section, we show that the derived mathematical model reduces to a conventional mathematical model of bubble expansion with uniform interior, with further crucial assumptions, such that evaporating velocity of liquid is sufficiently small compared with the liquid velocity at the bubble wall. We do not use the boundary conditions at the bubble wall, Eqs. (5.158) and (5.159) which contain C 4∗ and d4∗ . Consequently, the reduced model is less accurate in dealing with mass flux at the bubble wall. Although this model is still hard to solve analytically, we can at least discuss the asymptotic behavior of bubble motion, later in this chapter.

5.9.1 Assumptions We further impose the following assumptions: (i) The quantities ρV , pV , TV , and eV are uniform in the bubble; hence these quantities are functions of time alone, (ii) Evaporating velocity of liquid: j/ρ L is sufficiently small compared with the liquid velocity at the bubble wall w L , defined by Eq. (5.131); hence w L is equated ˙ Then, we replace Eq. (5.111) by to bubble wall velocity R. vL =

R 2 R˙ . r2

(5.293)

200

5 Dynamics of Spherical Vapor Bubble ˙ wL=R TVw = TLw

Vapor

Liquid

Temperature Distribution

Bubble Wall

Fig. 5.7 The velocity of bubble wall and the temperature distribution in the simplified model

The governing equation of bubble dynamics (5.147) becomes ρL

R R¨ +

3 ˙2 R 2

= pV (t) − p∞ (t) −

2σ , R

(5.294)

where liquid viscosity μ L is neglected for simplicity, (iii) Temperature discontinuity at the bubble wall written in Eq. (5.144) is negligible. Then, at the bubble wall, we impose TL |r =R = TV .

(5.295)

With the use of above assumptions, the schematic diagram of the model shown in Fig. 5.5 can be greatly simplified as shown in Fig. 5.7

5.9.2 Governing Equations and Conditions Now, we consider how other quantities depend on r by examining the conservation of mass for the spherically symmetric flows, Eq. (5.95), using the fact that ρV is independent of r , ∂ρV 2ρV vV ∂ = − (ρV vV ) − = −ρV ∂t ∂r r

∂vV 2vV + ∂r r

.

(5.296)

Notice that the left-hand side of Eq. (5.296) is independent of r , so is the right-hand side. Let ∂vV 2vV + = C(t), ∂r r

(5.297)

where C(t) is independent of r , i.e. a function of time only. The dependence of vV on r is easily found that vV = r D(t): under the condition that vV is bounded at r = 0, where D(t) is another function of time only; and then we have

5.9

Bubble Expansion with Uniform Interior

201

∂vV vV = . ∂r r

(5.298)

Substituting Eq. (5.298) into Eq. (5.296) we have, vV = −

r dρV . 3ρV dt

(5.299)

We only have to solve the equations for the outside of the bubble, i.e., those for liquid, due to the assumption of the uniformity inside the bubble. Since vcL becomes zero with the assumption (ii) in Sect. 5.9.1 and Eq. (5.174), the heat equation (5.180) can be further reduced ∂ 2 TL ∂ TL , = DL ∂s ∂z 2

(5.300)

where D L is the coefficient of thermal diffusivity of liquid and defined by DL =

λL . ρL cL

(5.301)

We now consider the boundary conditions at r = R. With the use of Eq. (5.299), the conservation equation of mass at the bubble wall (5.128) becomes ρV R˙ − w L = ρL

R dρV + R˙ 3ρV dt

d 1 = 2 4πρ L R dt

4 π R 3 ρV 3

.

(5.302)

The assumption (ii) in Sect. 5.9.1 implies that the order of magnitude of the righthand side of Eq. (5.302) is assumed to be sufficiently small compared with that of ˙ R. As we have discussed in Sect. 5.4.3, the latent heat term usually dominates over the other terms in the right-hand side of the conservation equation of energy at the bubble wall (5.141); hence heat transport to a whole bubble due to the evaporation is written as Ldm = Ld(4π R 3 ρV /3)/dt, where dm is an infinitesimal increment of mass of vapor in the bubble. On the other hand, heat conduction at the bubble wall from the external region dominates in the left-hand side of Eq. (5.141); hence heat transport from the external region to a whole bubble due to heat conduction is written as 4π R 2 λ L (∂ TL /∂r ). Equating these two heat transports, we obtain L d ∂ TL = λL ∂r 4π R 2 dt

4 π R 3 ρV 3

.

(5.303)

With the use of Eq. (5.264), Eq. (5.303) is also put in Lagrangian coordinates [cf. Eq. (5.164)]:

202

5 Dynamics of Spherical Vapor Bubble

λ

∂T L d = ∂z 4π R 4 dt

4 π R 3 ρV 3

=

L d 4π ds

4 π R 3 ρV 3

.

(5.304)

Now we seek for the analytical solution of Eqs. (5.294) and (5.300), following Plesset and Zwick [17, 25] with the assumption that the thermal diffusion length is usually small in comparison with the bubble radius for this class of problems; we solve the thermal problem based on the assumption of thin thermal boundary layer. This expansion is useful since the determination of the motion of the bubble wall depends essentially on the temperature variation in the neighborhood of the wall. In the following, subscript L is dropped for simplicity.

5.9.3 Heat Equation for Liquid We introduce new variables θ and U defined by θ (z, s) =

∂U = T − T∞ , ∂z

(5.305)

where T∞ = T (∞, 0). Introducing U enables us to solve the heat equation even when D in Eq. (5.301) is a function of r , although it is assumed constant in this analysis. Substituting Eq. (5.305) into Eq. (5.300) leads to the heat conduction equation: ∂U ∂U = D 2, ∂s ∂z

(5.306)

with boundary conditions: ∂θ (0, s) ∂ 2 U (0, s) L d =λ λ = 2 ∂z 4π ds ∂z ∂U (∞, s) = θ (∞, s) = 0, ∂z

4 π R 3 ρV 3

,

(5.307) (5.308)

and initial condition: θ (z, 0) = g(z).

(5.309)

The temperature at the bubble wall in Eq. (5.295) is defined by ∂U (0, s) = θ (0, s) = TV − T∞ . ∂z

(5.310)

Notice that neither R, ρV in Eq. (5.307) nor TV in Eq. (5.310) are not a priori determined, but they are parts of the solution. Function g(z) in Eq. (5.309) is an initial temperature distribution in the liquid. Suppose g(z) is given by

5.9

Bubble Expansion with Uniform Interior

203

g(z) = [TV (0) − T∞ ] e−z/z 0 ,

(5.311)

where z 0 is so chosen that a variable l, 1/3

l = z0 ,

(5.312)

is the thickness of the initial thermal boundary layer. This statement will be made clear in the following. As the liquid has been assumed incompressible, using Eq. (5.164), we have z(r, t) =

r

ξ 2 dξ =

R(t)

1 3 r − R(t)3 . 3

(5.313)

Substituting Eqs. (5.311) and (5.313) into Eq. (5.305), we have

(r 3 − R03 ) . T (r, 0) = T∞ + [TV (0) − T∞ ] exp − 3l 3

(5.314)

We integrate Eq. (5.311) once with respect to z and impose that U (∞, s) = 0, to obtain initial condition for U : U (z, 0) = −z 0 [TV (0) − T∞ ] e−z/z 0 .

(5.315)

5.9.4 Solution of Heat Equation Equation (5.306) with conditions (5.307), (5.308), and (5.315) can be solved using the Laplace transform: U(z, σ ) = L {U (z, s)} =

∞

e−σ τ U (z, τ )dτ.

(5.316)

0

Applying the Laplace transform, Eqs. (5.306), (5.307), and (5.308) become σ U(z, σ ) − U (z, 0) = D

∂2 U(z, σ ), ∂z 2

% & L d ! 3 " ∂ 2 U(0, σ ) { = J (σ ) = L j (s)} = L ρ , R V ∂z 2 3λ ds ∂U(∞, σ ) = 0. ∂z

(5.317)

(5.318)

(5.319)

204

5 Dynamics of Spherical Vapor Bubble

The solution which is bounded at r → ∞ for the ordinary differential equation (5.317) is given by β D σ J (σ ) − 2 exp −z , U(z, σ ) = β exp (−z/z 0 ) + s D z0

(5.320)

where β is defined as β=−

TV (0) − T∞ σ−

D z 02

.

(5.321)

Here, noticing Eq. (5.305), we should obtain the inverse Laplace transform of not U but ∂U/∂z; β ∂U(z, σ ) = − exp (−z/z 0 ) + ∂z z0

β D σ J (σ ) − 2 exp −z . σ D z0

(5.322)

With the use of formulas of inverse Laplace transform: ! ) "⎫ ⎧ σ ⎬ ⎨ exp −z D D z2 −1 ) L exp − = , ⎩ ⎭ σ πs 4Ds

(5.323)

D

⎧ ⎫ % & ⎨ TV (0) − T∞ ⎬ Ds β −1 −1 − = [TV (0) − T∞ ] exp L . − =L ⎩ z0 z 02 σ − D2 ⎭

(5.324)

z0

Suppose functions ϕ(s) and ψ(s) are given, then the convolution of these function ϕ(s) ∗ ψ(s) is defined as ϕ(s) ∗ ψ(s) =

s

ϕ(s − τ )ψ(τ )dτ.

(5.325)

0

Then, we obtain, θ =

% & ∂U ∂U = L−1 ∂z ∂z

D z z2 exp − − = [TV (0) − T∞ ] exp exp − z0 πs 4Ds TV (0) − T∞ Ds L d 3 (R ρV ) + exp . (5.326) ∗ 3λ ds z0 z 02 Ds z 02

5.9

Bubble Expansion with Uniform Interior

205

With the use of Eqs. (5.312), (5.164), and (5.171), we obtain D Ds = 2 z0

*t 0

R(x)4 dx z 02

=

D

*t 0

R(x)4 dx , l6

r 3 − R(t)3 z = . z0 3l 3

(5.327) (5.328)

Substituting Eqs. (5.326), (5.327), and (5.328) into Eq. (5.305), we obtain

*t D 0 R(x)4 dx r 3 − R(t)3 T (r, t) = T∞ + [TV (0) − T∞ ] exp − + 3l 3 l6 ⎧ (2 ⎫ ' 3 ⎬ ⎨ r − R(t)3 DT s(t) 1 )* − exp − * s(t) ⎩ 36D s(t) π 0 R(y)4 dy ⎭ R(y)4 dy ξ ξ ⎧ ⎤⎫ ⎡ * s(t) 4 dy ⎬ ⎨ L 4 D R(y) [TV (0) − T∞ ] R(t) d 3 ξ ⎦ dξ, × exp ⎣ (R ρV ) + ⎭ ⎩ 3R(t)4 λ dt l3 l6 (5.329) where Eq. (5.171) is used. Now, we evaluate T at r = R, using Eq. (5.329): L d 3 D 1 T (R, t) = T∞ − ρ ) ∗ (R V π s(t) 3R(t)4 λ dt 9 : Ds(t) Ds(t) D 1 + [TV (0) − T∞ ] exp ∗ exp − . (5.330) l6 πl 6 s(t) l6

Notice that

1 Ds(t) ∗ exp s(t) l6 s(t) D 1 D(s(t) − ξ ) dξ exp = ξ πl 6 0 l6 Ds(t) Ds(t) = exp , erf l6 l6 D πl 6

then, Eq. (5.330) becomes

(5.331)

206

5 Dynamics of Spherical Vapor Bubble

T (R, t) = T∞ − c

u 0

d (yρ )dv u u dv√ V + [TV (0) − T∞ ] exp erfc , u u u−v 0 0 (5.332)

where we introduce new variables and constants: y=

R R0

3

α and u = 4 R0

t

R(y)4 dy,

(5.333)

αl 6 . D R04

(5.334)

0

and L R0 c= 3λ

αD π

and u 0 =

Thus the temperature field T (r, t) is completely determined from Eq. (5.329), if R(t) and ρV (t) are known. On the other hand, if we turn our attention to Eq. (5.332), the role of temperature field in analysis of solution is slightly different. Notice that the left-hand side of Eq. (5.332) is equated with the uniform temperature inside bubble, TV (t), by using temperature boundary condition (5.295); hence Eq. (5.332) is rewritten as TV (t) = T∞ − c

u 0

d (yρ )dv u u dv√ V erfc . (5.335) + [TV (0) − T∞ ] exp u0 u0 u−v

Then, R(t) may be determined from Eq. (5.335), since y and u in Eq. (5.333) are functions of R(t), if ρV (t), pV (t), and TV (t) are known. We know that ρV (t), pV (t), and TV (t) are not given a priori, in general, but given as the solution of Eq. (5.335) in conjunction with (5.329), the equation of state [cf. Eq. (5.116)], and the equation for the phase equilibrium that typically determines the equilibrium vapor pressure at given temperature.

5.9.5 Asymptotic Growth of Vapor Bubble Assuming that pV = pe (TV ), and hence ρe = ρV (TV ),

(5.336)

where pe and ρe are the equilibrium vapor pressure and density, respectively, we study the asymptotic growth of a vapor bubble in a superheated liquid as t → ∞, by simplifying Eq. (5.335), noticing that the last term in the right-hand side of Eq. (5.335) vanishes as t → ∞, since exp(y)erfc(y) → 0 as y → ∞. Then, Eq. (5.335) can be approximated to become,

5.9

Bubble Expansion with Uniform Interior

207 u

T0 − Tb ∼ cρe (Tb ) 0

d y dv , as t → ∞. √dv u−v

(5.337)

Here, we assumed that, for sufficiently large t, the temperature inside the bubble is regarded as constant, TV (t) = Tb ; hence density of bubble is also regarded as constant, ρV (t; TV ) = ρe (t; TV ) = ρe (Tb ). We consider the following equation:

u

0

d y dv √dv = G, u−v

(5.338)

where G=

3λ R0 Lρe (Tb )

π (T0 − Tb ) . α DT

(5.339)

Equation (5.338) belongs to the class of Abel’s integral equations [18] (see Appendix C at the end of this book). We simply substitute Eq. (5.338) into Eq. (C.28), noticing that G is constant, and we obtain sin 12 π d dy = du π du

u 0

G 2G d √ dv = u. 1/2 (u − v) π du

(5.340)

Then solution of Eq. (5.338) is given by y=

2G √ u. π

(5.341)

By substituting Eq. (5.341) into Eq. (5.337), we obtain,

1 6λ (T0 − Tb ) π D Lρe (Tb )

R3 ∼

t

R 4 (y)dy.

(5.342)

0

Here we are only interested in the asymptotic behavior of bubble radius R as t → ∞; hence we assume that R has the form of a power-low function of t, R ∼ t k . Then Eq. (5.342) becomes t

3k

∼

4k+1 1 1 6λ (T0 − Tb ) t 2 . π D Lρe (Tb ) 4k + 1

(5.343)

Equating exponents of t in both sides of Eq. (5.343), we obtain k = 1/2. By dividing both sides by t 2 , we have [25] R∼2

λ 3 (T0 − Tb )t 1/2 , π Lρe (Tb )

as t → ∞.

(5.344)

208

5 Dynamics of Spherical Vapor Bubble

5.9.6 Bubble Motion Coupled with Heat Conduction We now try to obtain the complete picture of the bubble motion. We need to couple Eq. (5.332) with Eq. (5.294). Notice that R = R0 y 1/3 , du/dt = α(R/r0 )4 = αy 4/3 and α R0 y 2/3 dy , R˙ = 3 du

(5.345)

the left-hand side of Eq. (5.294) becomes d 1 3 R R¨ + R˙ 2 = (R0 α)2 2 6 dy

% y

7/3

dy du

& ,

(5.346)

and the right-hand side of Eq. (5.294) becomes ( 2σ 2σ 1' 1 . pV (R(t), t) − p∞ (t) − = pV (R(t), t) − p∞ (t) − ρ R ρ p R0 y 1/3 (5.347) Equating Eqs. (5.346) and (5.347), we have 1 d 6 dy

y

1 d = 6 dy

7/3

dy du

2

y

7/3

+φ+ dy du

1 1 2σ 2 1/3 ρ R0 (α R0 ) y

2 +φ+

1 y 1/3

= 0,

(5.348)

where φ=

( R0 ' p∞ − pe (TV ) , 2σ

(5.349)

) and α = (2σ/ρ R02 ). Notice that Eqs. (5.332) and (5.348) are connected when the equilibrium vapor pressure is specified as a function of temperature, this fact which is pointed out by Plesset and Zwick [17]. They assume that, for superheats not too far above the boiling temperature Tb of the liquid at the external pressure p∞ , the vapor pressure may be approximated by a linear function of the temperature: pe (TV ) − p∞ = A(T − Tb ). ρ

(5.350)

References

209

Thus ⎧ u d (yρ )dv R0 ρ A ⎨ dv√ V φ(TV ) = − T∞ − Tb − c 2σ ⎩ u−v 0 & u u erfc . + [TV (0) − T∞ ] exp u0 u0

(5.351)

With the use of the following expressions; r0 ρ A 1 ρe (TV ) cρe (Tb ) = , , ξ= , b= TΔ 2σ ρV (Tb ) TΔ

(5.352)

Eq. (5.348) with Eq. (5.332) becomes 1 d 6 dy

y

7/3

dy du

2

u d (yρ )dv 1 T∞ − Tb dv√ V − 1/3 − b = TΔ y u−v 0 u u (TV (0) − T∞ ) exp . + erfc TΔ u0 u0

(5.353)

Boundary conditions for Eq. (5.353) are obtained from Eqs. (5.333) and (5.345) as y = 1,

dy =0 du

at u = 0.

(5.354)

Solving Eq. (5.353) with (5.354) is out of the scope of this book. Further discussion may be found in, e.g. Plesset and Zwick [17, 25].

References 1. G.K. Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, Cambridge, 1967) 2. C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, Singapore, 1978) 3. A.E. Beylich, Dynamics and thermodynamics of spherical vapour bubbles. VDIForshungsheft 630, 1–27 (1985) 4. R.B. Bird, W.E. Stewart, E.N. Lightfood, Transport Phenomena (Wiley, New York, NY, 1960) 5. R.S. Borden, A Course in Advanced Calculus (Dover Press, New York, NY, 1997) 6. E.B. Dusssan, On the difference between a bounding surface and a material surface. J. Fluid. Mech. 75, 609–623 (1976) 7. H. Goldstein, Classical Mechanics, 2nd edn. (Addison Wesley Publishing, London, 1980) 8. M. Gurin, An Introduction to Continuum Mechanics (Academic, Orlando, FL, 1981) 9. D.Y. Hsieh, Some analytical aspects of bubble dynamics. J. Basic Eng. 87, 991–1005 (1965) 10. JSME Data Book: Thermophysical Properties of Fluids (Japan Society of Mechanical Engineers, Tokyo, 1983)

210

5 Dynamics of Spherical Vapor Bubble

11. D. Kondepudi, I. Prigogine, Modern Thermodynamics: From Heat Engines to Dissipative Structures, 2nd edn. (Wiley, London, 1999) 12. L. Landau, E. Lifshitz, Fluid Mechanics, 2nd edn. (Pergamon Press, Oxford, 1987) 13. L. Landau, E. Lifshitz, Statistical Physics, 3rd edn. (Pergamon Press, Oxford, 1980) 14. M.J. Miksis, L. Ting, Nonlinear radial oscillations of a gas bubble including thermal effects. J. Acoust. Soc. Am. 79, 997–905 (1984) 15. M. Minnarert, Musicalair-bubbles and sounds of running water. Philos. Mag. 16, 235–248 (1933) 16. S. Osher, R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces (Springer, New York, NY, 2003) 17. M.S. Plesset, S.A. Zwick, The growth of vapor bubbles in superheated liquids. J. Appl. Phys. 25, 493–500 (1954) 18. D. Porter, D.S.G. Stirling, Integral Equations (Cambridge University Press, Cambridge, 1990) 19. A. Prosperetti, Boundary conditions at a liquid-vapor interface. Meccanica 14, 34–47 (1979) 20. J.S. Rowlinson, B. Widom, Molecular Theory of Capillarity (Clarendon Press, Oxford,1982) 21. I. Stackgold, Green’s Functions and Boundary Value Problems (Wiley, New York, NY, 1979) 22. T. Tanahashi, Continuum Mechanics, vol. 6, (Rikoh Tosho, Tokyo, 1988) 23. L. Trilling, The collapse and rebound of a gas bubble. J. Appl. Phys. 23, 14–17 (1952) 24. A.H. Zemanian, Generalized Integral Transformations (Dover, New York, NY, 1987) 25. S.A. Zwick, M.S. Plesset, On the dynamics of small vapor bubbles in liquids. J. Math. Phys. 33, 308–330 (1955)

Appendix A

Vectors, Tensors, and Their Notations

A.1 Scalar, Vector, and Tensor A physical quantity appears in this book is a scalar, vector, or tensor. A quantity expressed by a number, such as mass, volume, temperature, is a scalar. If a quantity a is expressed as a one-dimensional array of numbers, a = (a1 , a2 , . . . , a N ), where N is the dimension of the physical space on which a is considered, it is a vector, and ai (i = 1, 2, . . . , N ) is called the ith component of vector a. The number N is three in this book. Displacement, velocity, and force are vectors. If a and b are vectors, a linear combination αa + β b,

(α and β are scalars),

(A.1)

gives a vector. Sometimes, ai is used as a representative of vector a. A tensor is a multi-dimensional array of numbers. If P is an n-dimensional array of numbers, it is called an nth-order tensor. A scalar and a vector may be called a zeroth-order tensor and a first-order tensor, respectively. An example of a secondorder tensor is a stress tensor in continuum mechanics. A second-order tensor can be expressed by a matrix. It is important to notice that a tensor is a multilinear functional of tensors.1 For example, a second-order tensor P(a, b), which is a functional of two first-order tensors (vectors) a and b, satisfies the bilinearity relations P(a 1 + a2 , b) = P(a1 , b) + P(a2 , b),

(A.2)

P(a, b1 + b2 ) = P(a, b1 ) + P(a, b2 ),

(A.3)

P(αa, b) = P(a, αb) = α P(a, b),

(A.4)

where α is a scalar. Remember that a stress tensor gives a stress if two unit vectors are specified, one determines the normal direction of a surface and another does the component of the stress acting on the surface. 1A

functional is a mapping of some functions to a number.

S. Fujikawa et al., Vapor-Liquid Interfaces, Bubbles and Droplets, Heat and Mass C Springer-Verlag Berlin Heidelberg 2011 Transfer, DOI 10.1007/978-3-642-18038-5,

211

212

Appendix A

If a tensor (scalar or vector) is a function of time t and position x, it is called a tensor field (scalar field or vector field). Usually, they are assumed to be continuously differentiable with respect to t and x. In fluid dynamics, a pressure field, velocity field, and stress field are, respectively, a scalar field, vector field, and tensor field, and all of them are continuously differentiable. Thus, the basic conservation laws of physics can be expressed in the forms of partial differential equations consisted of scalars, vectors, and tensors. It is essential for physics and its applications to engineering that the basic conservation laws expressed by scalars, vectors, and tensors are unchanged under the rotation of coordinate system with a fixed origin and the Galilean transformation.

A.2 Einstein Summation Convention After a Cartesian coordinate system x = (x1 , x2 , x3 ) is specified, the components ai ’s (i = 1, 2, 3) of vector field a and the components Pi j ’s (i, j = 1, 2, 3) of a second-order tensor field P are determined.2 The expressions of mathematically complicated equations can often be made compact by using symbols like a and P instead of ai ’s and Pi j ’s. However, the numerical evaluations of vectors and tensors require the handling of their components. In the following, we summarize notations of some binary-product operations of vectors and tensors presented in component forms. The inner product (or scalar product) of a second-order tensor P and a vector a gives a vector b, i.e., b = P · a,

(A.5)

and this can be written as bi =

3

Pi j a j ,

(i = 1, 2, 3).

(A.6)

j=1

According to the Einstein summation convention, we can eliminate the summation symbol to yield bi = Pi j a j ,

(i = 1, 2, 3).

(A.7)

The Einstein summation convention is a rule of notation of binary-product operations of vectors and tensors in a single term, which states that if a same index (subscript) appears twice in a single term, then the summation is taken from one to

2 The

representations of vectors and tensors in component forms are possible in arbitrary curvilinear coordinate systems.

Appendix A

213

three for the index in the term. The index is called a dummy index. Hereafter, we use the Einstein summation convention. The scalar product of two vectors, a and b, gives a scalar α, α = a · b = ai bi .

(A.8)

The dyadic of two vectors gives a second-order tensor P, P = ab = ai b j = Pi j ,

(i, j = 1, 2, 3),

(A.9)

where, as usual, we do not distinguish a tensor P from its representative expression Pi j , although the notation Pi j is often used as the (i, j)th component of tensor P. The scalar product (or contraction) of two tensors T and U, denoted by T : U, gives a scalar α, α = T : U = Ti j Ui j .

(A.10)

The gradient of a scalar field f is a vector, and can be expressed as grad f = ∇ f =

∂f , ∂ xi

(i = 1, 2, 3).

(A.11)

The divergence of a vector field v is a scalar expressed as div v = ∇ · v =

∂vi . ∂ xi

(A.12)

The strain rate tensor3 in fluid dynamics, ε, can be constructed by the dyadic of vectors ∇ and v as4 1 ∂v ∂v j 1 i T = εi j , (i, j = 1, 2, 3), = ε= + (A.13) ∇v + (∇v) 2 2 ∂x j ∂ xi where the superscript T denotes the transpose of matrix. Here, to simplify the notation further, we can indicate the differentiation with respect to xi by index i after an index denoting a component of vector or tensor with a comma separating the two indices. That is, grad f = f ,i ,

3 It

div v = vi,i ,

ε=

$ 1# vi, j + v j,i , 2

(A.14)

is sometimes called the rate-of-strain tensor or rate-of-deformation tensor.

∇ is not a vector because it is not an array of numbers but an array of differential operators, ∇ = (∂/∂ x 1 , ∂/∂ x2 , ∂/∂ x3 ). 4 Precisely,

214

Appendix A

where since f is a scalar, no index appears before the comma before index i indicating the differentiation with respect to xi . The Einstein summation convention is also applied to this type of simplified notation as shown in the second equation in Eq. (A.14). The Kronecker delta δi j is a representation of the second-order identity tensor I, given by 9 δi j =

1 if i = j, 0 otherwise.

(A.15)

The identity transformation from a vector a to a vector b can be written with the Kronecker delta as b = I · a = δi j a j = ai ,

(i = 1, 2, 3).

(A.16)

The Eddington epsilon i jk defined by

i jk

⎧ ⎪ 1 (i, j, k) = (1, 2, 3), (2, 3, 1), (3, 1, 2), ⎪ ⎨ = −1 (i, j, k) = (3, 2, 1), (2, 1, 3), (1, 3, 2), ⎪ ⎪ ⎩ 0 i = j or j = k or k = i,

(A.17)

is the third-order alternating unit tensor. A vector product of two vectors and curl operation to vector field v can be expressed as c = a × b = i jk a j bk = ci , (i = 1, 2, 3), ∂vk = i jk vk, j , (i = 1, 2, 3). curl v = ∇ × v = i jk ∂x j

(A.18) (A.19)

Several relations involving δi j and i jk are useful in manipulations of vectors and tensors: δi j δi j = 3, i jk i jk = 6, i jk h jk = 2δi h , i jk mnk = δim δ jn − δin δ jm .

(A.20) (A.21) (A.22)

A second-order tensor P is called symmetric, if P = P T , or Pi j = P ji ,

(i, j = 1, 2, 3).

(A.23)

Clearly, the strain rate tensor ε and the Kronecker delta are the symmetric secondorder tensors.

Appendix B

Equations in Fluid Dynamics

B.1 Conservation Equations Let the macroscopic variables be defined everywhere in a space filled with a fluid, and let them be continuously differentiable functions of time t and position x. The macroscopic variables that should be defined at this stage are the density ρ, velocity v, internal energy per unit mass e, stress tensor P, and heat flux q. Then, the conservation equations of mass, momentum and energy of the fluid, in general, are respectively written as ∂ρ + ∇ · (ρv) = 0, ∂t

(B.1)

∂ρv + ∇ · (ρvv + P) = ρb, (B.2) ∂t 1 2 ∂ 1 2 ρ v + ρe + ∇ · ρ v + ρe v + v · P + q ∂t 2 2 = ρb · v + ρ S, (B.3) where b is a body force exerted on the fluid per unit mass, 1 2 ρ v + ρe 2

(B.4)

is the total energy of the fluid per unit volume, and S is a heat generated in the fluid per unit mass and per unit time. The body force b and heat generation S are independent of the motion of fluid and prescribed by some other rules. Equations (B.1), (B.2), and (B.3) are the most fundamental equations in fluid dynamics, and can be derived, for example, by considering the conservation of mass, momentum, and energy in a volume element in the physical space without specifying the explicit forms of P and q. Furthermore, the relation between the density ρ and the internal energy e is not necessary for the derivation of Eqs. (B.1), 215

216

Appendix B

(B.2), and (B.3).1 Clearly, the number of unknown variables in Eqs. (B.1), (B.2), and (B.3) exceeds the number of Eqs. (B.1), (B.2), and (B.3), and therefore we have to add some equations. Usually, fluid dynamics assumes that (1) The fluid is a Newtonian fluid in the sense that the stress tensor is given by the sum of the pressure p and the viscous stress tensor τ ,2 P = pI − τ, 2μ τ = 2με + μb − (ε : I)I, 3

(B.5) (B.6)

where μ is the viscosity coefficient, μb is the bulk viscosity coefficient,3 ε is the strain rate tensor defined by Eq. (A.13) in Appendix A, and the operator : means the contraction of two second-order tensors defined by Eq. (A.10) in Appendix A. Since the strain rate tensor ε and the identity tensor are symmetric, the viscous stress tensor τ is also symmetric. The viscosity coefficients are usually assumed as functions of temperature and pressure.4 (2) The heat flux obeys the Fourier law, q = −λ∇T,

(B.7)

where λ is the thermal conductivity coefficient and T is the temperature of fluid. The thermal conductivity coefficient is usually assumed as a function of temperature and pressure. (3) The thermodynamic relations hold among the pressure p, temperature T , internal energy e, and density ρ. This is the assumption of local equilibrium state. For the above four thermodynamic variables, there exist two independent thermodynamic relations. For example, if the fluid is an ideal gas, we have p = ρ RT,

e = cv T,

(B.8)

where the first one is the (thermal) equation of state of ideal gas (R = k/m is the gas constant, k is the Boltzmann constant, and m is a mass of a molecule) and the second is the (caloric) equation of state of ideal gas (cv is the specific heat for constant volume per unit mass). If the gas is treated as an incompressible fluid, the density ρ is not a thermodynamic variable. Then, the first equation in Eq. (B.8) should be discarded and the definition of incompressible flows

1 In the incompressible fluid flows, we cannot assume any relation between ρ

and other thermodynamics variables. Nevertheless, the conservation laws (B.1), (B.2), and (B.3) should be satisfied.

2 In

many textbooks of fluid dynamics, the sign of stress tensor P is opposite to Eq. (B.5).

3 The 4 The

bulk viscosity coefficient is sometimes called the second viscosity coefficient.

viscosity coefficients and thermal conductivity coefficient of an ideal gas are functions of temperature.

Appendix B

217

∂ρ + v · ∇ρ = 0, ∂t

(B.9)

should be used instead. At least for ideal gases, the above three statements are theoretically validated by the kinetic theory of gases in the limit that the Knudsen number goes to zero, if the nonlinearity is sufficiently weak.5 For liquids, although there are no theoretical validations for Eqs. (B.5), (B.6), and (B.7), they are as a whole admitted and significant objections have never been raised against them.6 Thus, the system of equations in fluid dynamics is closed. In principle, we can solve it under appropriate boundary conditions and initial condition. The set of equations, Eqs. (B.1), (B.2), and (B.3) with Eqs. (B.5), (B.6), and (B.7) may be called the set of Navier–Stokes equations.7 Equations (B.1), (B.2), and (B.3) are written in the so-called conservation law form, ∂ (ρ f ) = −∇ · (ρ f v + φ) + ρϑ, ∂t

(B.10)

where f and ϑ are vectors or scalars and φ is a vector or a tensor. In fact, Eqs. (B.1), (B.2), and (B.3) are recovered as follows: Eq. (B.1) :

f = 1,

φ = 0,

Eq. (B.2) :

f = v,

φ = P,

Eq. (B.3) :

f =

1 2 v + e, 2

ϑ = 0,

(B.11)

ϑ = b, φ = v · P + q,

(B.12) ϑ = b · v + S.

(B.13)

In the above three equations, ρ f v+φ is very important for understanding the physics related to the interface: ρv in Eq. (B.1) is called the mass flux density vector, ρvv + P in Eq. (B.2) is called the momentum flux density tensor, and ρ( 12 |v|2 + e)v + v · P + q in Eq. (B.3) is called the energy flux density vector. In fluid dynamics, in addition to Eq. (B.3), there are several variations in the equation associated with the energy. For example, the equation of the internal energy 5 See

Footnotes 20 and 21 in Chap. 2. fluids are excluded, of course.

6 Non-Newtonian

7 The name “Navier–Stokes equations” is often used to indicate the momentum conservation equa-

tions Eq. (B.2) with the stress tensor of Newtonian fluid (B.5) and (B.6) or its variations, ρ

∂v = −ρ(v · ∇)v − ∇ p + ∇ · τ + ρb, ∂t

∂v 1 + (v · ∇)v = −∇ p + μ∇ 2 v + μb + μ ∇ (∇ · v) + ρb, ∂t 3 for constant μ and μb .

and

ρ

218

Appendix B

per unit volume can be written as ∂ρe = −∇ · (ρev) − p∇ · v + τ : ε − ∇ · q + ρ S. ∂t

(B.14)

B.2 Conservation Equations in Component Forms As mentioned in Appendix A, actual numerical evaluations of vectors and tensors require the handling of their components. We therefore write down Eqs. (B.1), (B.2), and (B.3) and Eqs. (B.5), (B.6), and (B.7) in component forms with indices using the Einstein summation convention explained in Appendix A. The mass conservation equation (B.1): ∂ρvi ∂ρ = 0. + ∂t ∂ xi

(B.15)

The momentum conservation equation (B.2): ∂ρvi v j + Pi j ∂ρvi = ρbi , + ∂t ∂x j

(i = 1, 2, 3).

(B.16)

The stress tensor of Newtonian fluid (B.5) and (B.6): Pi j = pδi j − τi j , ∂v j ∂vi 2μ ∂vk τi j = μ + μb − + δi j , ∂x j ∂ xi 3 ∂ xk

(B.17) (B.18)

where i, j = 1, 2, 3. The energy conservation equation (B.3): 1 2 ∂ 1 2 ρvi + ρe + ρvi + ρe v j + vi Pi j + q j = ρb j v j + ρ S. 2 ∂x j 2 (B.19) The heat flux based on the Fourier law (B.7): ∂ ∂t

q j = −λ

∂T , ∂x j

( j = 1, 2, 3).

(B.20)

Appendix C

Supplements to Chapter 5

C.1 Generalized Stokes Theorem We here prove the generalized Stokes theorem by using the Gauss theorem: .

∇ · WdV = V

W · nd S,

.

or

S

V

∂n W..n d V =

S

W..i n i d S,

(C.1)

where W (W..n ) is a tensorial quantity of any order. The Gauss theorem turns the surface integral of W over a closed surface S which is enclosing a volume V into the volume integral of a derivative of W (the divergence) over the interior of S, i.e., over the volume of V . We assume here that the surface S where the integration is evaluated is the plane surface as shown in Fig. C.1, for simplicity. Although this choice of the integral surface is rather special, the following discussion is also valid for general integral surfaces by considering the integration over an infinitesimal area element on the tangential surface at a point of contact. nT = n ST SS

n

nS = nC

S

nC nB = – n

tC

h

C SB

Fig. C.1 A volume considered in the proof of the generalized Stokes theorem

We draw a smooth closed line C on the plane surface S, and construct a column perpendicular to its base whose peripheral edge is C, as shown in Fig. C.1. The 219

220

Appendix C

height of this column is h. The unit normal vector to the plane surface S is denoted as n. The column is enclosed by the lateral closed surface S S and the top and bottom base surfaces S T and S B . The unit normal vectors to these three surfaces are n S , nT , and n B , respectively. The unit normal and tangential vectors to the closed line C are defined as nC and t C , respectively. Now we substitute T × n (nml T..m nl ) into W of Eq. (C.1) to obtain V

. ∇ · (T × n)d V = (T × n) · n S d S S S S + (T × n) · nT d S T + (T × n) · n B d S B , ST

(C.2)

SB

where T is a tensor of any order. The integrand of the left-hand side of Eq. (C.2) can be rewritten as

h

∇ · (T × n)d V =

. (∇ × T ) · nd Sdh,

0

V

(C.3)

S

by using the fact that n is constant since the surface S is plane. Now we rewrite the right-hand side of Eq. (C.2). Notice that the following holds: (T × n) · n = i jk T.. j n k n i = [n 1 (T..2 n 3 − T..3 n 2 ) + n 2 (T..3 n 1 − T..1 n 3 ) + n 3 (T..1 n 2 − T..2 n 1 )] = [T..1 (n 2 n 3 − n 3 n 2 ) + T..2 (n 3 n 1 − n 1 n 3 ) + T..3 (n 1 n 2 − n 2 n 1 )] = 0, (C.4) and that n T is equal to n and n B is equal to −n. Therefore only the integration over the lateral surface of the column contributes to Eq. (C.2). Since n S is written as nC on the lateral surface, the left-hand side of Eq. (C.2) is rewritten as

. SS

(T × n) · n S d S, = 0

h

. (T × n) · nC dldh.

(C.5)

C

With the use of nC = t C × n and n · t C = 0, the integrand of the left-hand side of Eq. (C.5) can be rewritten as ! " (T × n) · nC = (T × n) · t C × n = i jk T.. j n k imn tmC n n # $ = δ jm δkn − δ jn δkm T.. j n k tmC n n = T.. j n k t Cj n k − T.. j n k tkC n j " ! ! " = T · t C (n · n) − (T · n) t C · n = T · t C . (C.6) So we have the right-hand side of Eq. (C.2) as

Appendix C

221

.

SS

.

h

S

(T × n) · n d S, =

T · t C dldh.

0

(C.7)

C

Equating Eqs. (C.3) and (C.7), the Gauss theorem is rewritten as

h 0

.

h

(∇ × T ) · nd Sdh =

.

0

S

T · t C dldh.

(C.8)

C

Equation (C.8) holds for arbitrary choice of h, and therefore we finally obtain . (∇ × T ) · nd S = T · t C dl. (C.9) S

C

C.2 Characteristic Time of Heat Conduction We discuss the characteristic time of heat conduction by considering the simplest case, i.e., heat conduction in a uniform rod. Temperature u at position x in a uniform rod is governed by one-dimensional heat conduction equation: ∂ 2u ∂u = D 2, ∂t ∂x

(D > 0),

(C.10)

where D is the coefficient of thermal diffusivity. We first investigate the case that the rod is infinitely long; hence the domain of definition is −∞ < x < ∞. Suppose that temperature distribution at t = 0 is given as u|t=0 = ϕ(x),

(−∞ < x < ∞).

(C.11)

It is easily verified that solution of Eq. (C.10) is written as: u(x, t) =

∞

−∞

! " exp −Dλ2 t [A(λ) cos λx + B(λ) sin λx] dλ,

(C.12)

with coefficients A and B to be determined using initial condition (C.11). The substitution of the formal solution (C.12) into the initial condition (C.11) provides the Fourier Integral representation of ϕ(x): ϕ(x) =

∞ −∞

[A(λ) cos λx + B(λ) sin λx] dλ.

(C.13)

Coefficients A(λ) and B(λ) in Eq. (C.13) are obtained as A(λ) =

1 2π

∞

−∞

ϕ(ξ ) cos λξ dξ, B(λ) =

1 2π

∞

−∞

ϕ(ξ ) sin λξ dξ.

(C.14)

222

Appendix C

Substitution of relation (C.14) into Eq. (C.13) provides the following representation of u(x, t): u(x, t) =

1 2π

∞

−∞ ∞

dλ

∞ −∞

! " exp −Dλ2 t ϕ(ξ ) cos λ(x − ξ )dξ

∞ ! " 1 = ϕ(ξ )dξ exp −Dλ2 t cos λ(x − ξ )dλ 2π −∞ −∞ ∞ 1 (ξ − x)2 ϕ(ξ ) √ dξ. exp − = 4Dt 2 π Dt −∞

(C.15)

Now we solve the heat conduction problem in a semi-infinite rod. The governing equation is Eq. (C.10). We consider that the initial temperature of the rod is uniform and the temperature of an end of a rod is set as 0. Then, the boundary and initial conditions are written as u(0, t) = 0,

(C.16)

lim u(x, t) = 1,

(C.17)

⎧ ⎨ 1 if x > 0, 0 if x = 0, u(x, 0) = ϕ(x) = ⎩ −1 if x < 0,

(C.18)

x→∞

where all variables have been nondimensionalized. We require that ϕ(x) should be an odd function for Eq. (C.17) to be satisfied. This can be easily shown as follows. First we divide the solution (C.15) into two parts and rewrite as (ξ − x)2 ϕ(ξ ) exp − u(x, t) = √ dξ 4Dt 2 π Dt 0 & ∞ (ξ + x)2 + ϕ(−ξ ) exp − dξ . 4Dt 0 1

%

∞

(C.19)

Now we seek for the condition on which the solution satisfies Eq. (C.17) to be imposed on ϕ(x). We have from Eq. (C.19), 1 u(0, t) = √ 2 π Dt

∞ 0

ξ2 [ϕ(ξ ) + ϕ(−ξ )] exp − dξ = 0. 4Dt

(C.20)

We find that Eq. (C.20) holds if and only if the relation ϕ(−ξ ) = −ϕ(ξ ),

(C.21)

Appendix C

223

is satisfied; hence ϕ(ξ ) should be an odd function. √ Substituting √ Eq. (C.18) in (C.19), and Setting σ1 = (ξ − x)/(2 Dt) and σ2 = (ξ + x)/(2 Dt), the solution is given by 1 u(x, t) = √ 2 π Dt 1 = √ π 2 = √ π

∞

χ

& (ξ − x)2 (ξ + x)2 exp − − exp − dξ 4Dt 4Dt

0

−χ

∞%

" ! 2 exp −σ1 dσ1 −

∞ χ

" ! 2 exp −σ2 dσ2

" ! exp −σ 2 dσ = erf(χ ),

(C.22)

0

where x χ= √ . 2 Dt

(C.23)

Therefore the solution of the one-dimensional heat conduction equation (C.10) can be written by using only χ defined by (C.23). In Sect. 5.5, we use this χ to discuss the characteristic time of heat conduction.

C.3 Abel’s Integral Equation We shall seek the solution x of Abel’s integral equation,

t

f (t) = 0

x(η) dη (t − η)ν

(0 < ν < 1),

(C.24)

where f (t) is a given continuously differentiable function. Introducing a function φ(t) defined by φ(t) =

t

x(η)dη,

(C.25)

0

and using a formula π = sin νπ

t η

dξ (t

− ξ )1−ν (ξ

− η)ν

,

(C.26)

224

Appendix C

we can easily carry out the following integrations: π φ(t) = sin νπ

t

dη

t

=

x(η) dξ (t − ξ )1−ν (ξ − η)ν

η

0

t

ξ

x(η) dη (t − ξ )1−ν (ξ − η)ν

dξ 0

= 0

0 t

f (ξ ) dξ, (t − ξ )1−ν

(C.27)

where Eq. (C.24) has been used in the last equation of Eq. (C.27). Differentiating Eq. (C.27) with respect to t gives the solution of Eq. (C.24) as x(t) =

sin νπ d π dt

0

t

f (ξ ) dξ. (t − ξ )1−ν

(C.28)

Index

A Abel’s integral equation, 207, 223 Adsorbed liquid film, 78 Antoine’s equation, 8, 80 Association, 88 degree of, 88 B Berthelot equation, 138 BKW equation, 42 Boltzmann constant, 14, 20, 32, 113 Boltzmann equation, 1, 39 Bubble wall, 168, 200 Bulk viscosity, 57, 216 C Clausius–Clapeyron equation, 127 Coefficient of thermal diffusivity, 174, 201, 221 Collision frequency, 40 Collision term, 39, 41 Complete-condensation condition, 44 Compression factor, 2, 116 Condensation coefficient, 4, 44, 53, 64 Condensation mass flux, 72 Conservation equation of energy at bubble wall, 171, 175, 201 for spherical bubble, 164 on interface, 161 Conservation equation of mass at bubble wall, 168, 201 for spherical bubble, 163 on interface, 158 Conservation equation of momentum at bubble wall, 170 for spherical bubble, 164 on interface, 160 Conservation equations, 25, 153, 215 Convective derivative, 149, 151

Convolution, 183, 204 Critical temperature, 112 Cut-off radius, 34, 113 D Density distribution function, 23 Diffuse-reflection condition, 43 Dirac delta function, 25, 182 E Einstein summation convention, 39, 85, 160, 212 Energy flux density vector, 217 Energy reflectance, 89 Equipartition theorem, 20 Error function, 10, 184, 205 Euler equations, 54, 55 Evaporation coefficient, 4, 44, 50, 64, 131 Evaporation into vacuum, 46, 132 F Flux balance on interface, 157 Fourier law, 165, 170, 216, 218 G Gauss divergence theorem, 26, 145, 147, 150, 151 Gaussian–BGK Boltzmann equation, 5, 55, 77 Generalized Stokes theorem, 160, 221 Ghost effect, 54 Gibbs dividing surface (equimolor dividing surface), 129 Grad–Boltzmann limit, 41 H Half-space problem, 55, 62 Hamilton’s canonical equations of motion, 22 Hamiltonian, 22, 34, 37 Heat conduction, 174

225

226 Heat equation, 178, 194, 202, 221 Heat flux, 29 Heaviside function, 179 Helmholtz free energy, 153 Hertz–Knudsen–Langmuir formula, 66, 132 H-theorem, 42, 56 I Ideal gas, 20, 41 Incident shock wave, 79 Incompressible fluid, 216 Interface, 3 Interface velocity, 145 Interferometer, 89 Intermolecular force, 21 Internal degrees of freedom, 51, 56, 74 Intramolecular force, 21 Inverse Laplace transform, 181 K Kelvin equation, 111, 127, 169 Kinetic boundary condition, 3, 43, 50, 53, 57 Kinetic theory, 38 Kinetic theory of gases, 25, 38 Knudsen layer, 3, 55, 61 Knudsen layer analysis, 61 Knudsen number, 2, 55 L Laplace equation, 126 Laplace transform, 179, 203 Latent heat, 171, 201 Leap-frog scheme, 33, 113 Lennard-Jones potential, 31, 113 Liouville equation, 24 Liquid film, 79 Liquid temperature at bubble wall, 188 gradient at bubble wall, 189 of bubble interior, 186 Liquid velocity at bubble wall, 165, 169 Local equilibrium, 20, 28, 38, 55, 59 Local Maxwellian, 41 Loschmidt constant, 20 M Mach number, 82 Mass flux across the interface, 45, 47, 53, 65, 158 of molecules spontaneously evaporating, 47, 72, 131 Mass flux density vector, 217 Mass fraction, 87

Index Maxwell distribution function, 40 Mean collision frequency, 40–41 Mean free path, 3, 40–42, 57, 116 Mean free time, 14, 40 Molecular dynamics, 31, 46 Molecular gas dynamics, 25 Momentum flux density tensor, 217 Monatomic molecule, 21, 39, 72 Moving boundary problem, 176 N Nanodroplet, 112 Navier–Stokes equations, 1, 54–55, 217 Net mass flux of condensation, 78 Net mass flux of evaporation, 76 Newton’s equation of motion, 21 Newtonian fluid, 216, 218 Noncondensable gas, 87 Nonequilibrium state, 2, 46, 78, 131 N V E simulation, 31, 113 P Partition function, 74 Periodic boundary condition, 35, 113 Permanently absorbed liquid film, 93 Phase space, 23, 25 Polyatomic molecule, 50, 55, 73 Prandtl number, 42, 57, 192 Pressure of bubble interior, 196 R Ratio of specific heats, 56, 75 Rayleigh–Plesset equation, 172 Reflected shock wave, 79 Reflection mass flux, 72 S S expansion, 58 Saturated vapor density, 8, 44, 72 Saturated vapor pressure, 8, 116 Schrage formula, 66 Second harmonics, 108 Shock tube, 6, 78 Shock wave, 6, 13, 78 Slip coefficient, 62, 63 Solvability condition, 59 Sound resonance, 107 Sound resonance method, 106 Standing wave, 108 State equation of real gas, 138 Stress tensor, 28, 29 Surface entropy, 152 Surface of tension, 129 Surface tension, 116, 124, 159

Index T Temperature discontinuity at bubble wall, 158, 171 Temporal transition phenomenon, 12 Temporarily adsorbed liquid film, 94 Thermal diffusion-controlled condensation, 13 Thickness of interface, 2, 46, 129 Tolman equation, 111, 130 Tolman length, 129 Total curvature, 147 Transition layer, 2, 35, 116 Transition time, 11 Triple point temperature, 36, 50, 112, 138 U Unit normal of interface, 145

227 V Vapor pressure, 116 Vapor temperature at bubble wall, 195 gradient at bubble wall, 195 of bubble exterior, 194 outside thermal boundary layer, 190 with uniform interior, 206 Vapor velocity at bubble wall, 165, 169 Vapor–liquid interface, 143 Velocity distribution function, 39, 48, 73, 77 Velocity field of bubble interior, 197 Velocity scaling, 47, 114, 133 Volterra integral equation of the second kind, 9, 80